H^{2}\right)\left(u_{a}u_{b}-n_{a}n_{b}+p_{ab}\right)$ (5.195)

and equation (5.194) gives

$\rho=\frac{c}{A_{1}^{2}(t)A_{2}^{2}(t)}.$ (5.196)

Therefore, in the Bianchi I spacetime, we know the string fluid as well as the
magnetic field.

####  5.9 Conclusions

In this chapter, we have applied the 1+1+2 decomposition to the case of the
EMSF in the RMHD approximation. We have shown that a geometric assumption in
the form of a symmetry effects both the kinematics and the dynamics of the
resulting EMSF. We have approached the problem in two steps: a. In full
generality independently of a particular symmetry; and b. In the case of a CKV
which is either of the form $\xi^{a}=\xi u^{a}$ or of the form $\xi^{a}=\xi
n^{a}$ with $n^{a}=\frac{H^{a}}{H}$, where $u^{a}$ is the four-velocity of the
fluid and $H^{a}$ is the magnetic field. We applied the results of the case
$\xi^{a}=\xi u^{a}$ in the FRW spacetime, and the results of the case
$\xi^{a}=\xi n^{a}$ in the Bianchi I spacetime. In the latter case, we found
new solutions for the gravitational field.

It is apparent that the results stated in this chapter due to their generality
can be used in many different situations involving the electromagnetic field
and various types of symmetries. However, one may ask, if all the initial
conditions are viable for the solutions which follow from the existence of
symmetries. In particular, the existence of a symmetry in a solution is a
strong argument which when it is violated leads to other kinds of solutions.
On the other hand, from the theory of similarity solutions of differential
equations [110, 111], we know that for a given differential equation a
similarity solution satisfies the initial value problem/boundary conditions
iff the later are also invariant under the action of the symmetries which
provide the similarity transformations. This property can be applied in order
to define initial conditions for which a nonsymmetric solution can be related
with a symmetric one. For instance, to relate the inner and outer solutions in
a compact body.

###  Chapter 6 Constructing the CKVs of Bianchi III and V spacetimes

####  6.1 Introduction

The knowledge of the proper CKVs (see sec. 2.2) of a given spacetime is
important because they act as geometric constraints which can be used in the
study of the kinematics and the dynamics of a given spacetime. For example, a
CKV can be used to reduce the number of unknowns of a gravitational (or
cosmological) model and, also, to increase the possibility of finding new
solutions of Einstein’s field equations (see e.g. [106, 112, 113, 114, 115,
116, 117, 118] and chapter 5). Furthermore, the conformal algebra can be used
to classify spaces (e.g. Finsler manifolds and pseudo-Euclidean manifolds)
[119, 120]. For example, one may use the CKVs of a space in order to determine
the classes of manifolds which are conformally related to the given space; or
in order to study the locally conformal flatness of a space around a
singularity (i.e. a point $x_{0}$ where the CKV vanishes). The special class
of the CKVs, the KVs, have been used in numerous applications. For example,
Geroch in [121] and [122] has used the KVs in order to derive new solutions of
the gravitational field equations. The gradient KVs can be used also to
decompose the spacetime metric and to simplify the field equations. Moreover,
the KVs are related with the conservation laws for the geodesic equations.
Indeed, the KVs form a subalgebra on the Noether symmetries for the geodesic
Lagrangian [123].

Apart from the above applications, another important area where the CKVs and
the more general PCs have been used is the geometric study of Lie symmetries
of differential equations. For example, early studies of the geodesic
equations [62, 124, 125, 126] have shown a unique connection of the Lie point
symmetries of the geodesic equations in a Riemannian space with the elements
of the projective algebra of this space. Furthermore, in a conservative
dynamical system one may consider the kinetic energy as a metric (i.e. the
kinetic metric) and study the Lie and the Noether point symmetries of the
dynamical equations using the collineations of this metric. In [127], it has
been shown that in such systems, the Lie point symmetries are generated by the
special projective algebra and the Noether point symmetries by the Homothetic
algebra of the kinetic metric. Similar results have been found for some PDEs
of special interest in curved spacetimes, as the wave and the heat equation
(see e.g. [128, 129, 130] and references therein), where it has been shown
that the Lie point symmetries involve the CKVs.

In this chapter, we apply the propositions and the methodology developed in
[109, 131, 132] in order to determine all Bianchi III and V spacetimes that
admit proper CKVs. The Bianchi I spacetimes which admit proper CKVs have been
determined in111 For completeness, we restate these results in sec. 6.6 below.
[109]. Bianchi III and V spacetimes are of special interest and have many
applications in the study of anisotropic cosmologies. Bianchi spacetimes can
provide a different cosmological behavior in the early universe [133], while
also they can been seen as the homogeneous limits of exact inhomogeneous
cosmological models [134].

Bianchi spacetimes are spatially homogeneous spacetimes of the general form

$ds^{2}=-dt^{2}+A^{2}(t)(\omega_{1})^{2}+B^{2}(t)(\omega_{2})^{2}+C^{2}(t)(\omega_{3})^{2}$
(6.1)

where $\omega_{i}$, $i=1,2,3$, are basis 1-forms and $A(t)$, $B(t)$, $C(t)$
are functions of the time coordinate (see e.g. [135, 136, 137]). For instance:

Bianchi I $\displaystyle:$
$\displaystyle\omega_{1}=dx,~{}\omega_{2}=dy,~{}\omega_{3}=dz$ Bianchi III :
$\displaystyle\omega_{1}=dx,~{}\omega_{2}=dy,~{}\omega_{3}=e^{-x}dz$ Bianchi V
: $\displaystyle\omega_{1}=dx,~{}\omega_{2}=e^{x}dy,~{}\omega_{3}=e^{x}dz.$

In the case of $B^{2}(t)=C^{2}(t)$, the Bianchi spacetimes contain a fourth
isometry (the rotation of the $yz$-plane) and reduce to the important subclass
of Locally Rotational Symmetric (LRS) spacetimes (see e.g. [138] and citations
therein).

####  6.2 Preliminaries: Decomposable spacetimes

We recall that a Riemannian manifold is decomposable along the coordinate $t$
iff the metric $g_{ab}$ admits the non-null gradient KV $u^{a}=\partial_{t}$,
where $u^{a}u_{a}=\varepsilon u^{2}$ and $\varepsilon=\pm 1$. In this case,
one defines the projection operator

$h_{ab}=g_{ab}-\frac{\varepsilon}{u^{2}}u_{a}u_{b}$ (6.2)

and decomposes the tensor algebra along $u^{a}$ and normal to $u^{a}.$ For an
$n$-dimensional decomposable Riemannian manifolds $M^{n}$ with $n\geq 3$ an
algorithm has been developed [131] which determines the proper CKVs in terms
of the (gradient) proper CKVs of the $\left(n-1\right)$-dimensional non-
decomposable space.

In particular, it has been shown that _an $n$-dimensional decomposable space
$M^{n}$ admits proper CKVs iff the $\left(n-1\right)$-space $M^{n-1}$ admits a
gradient proper CKV whose conformal factor is the gradient factor which
constructs the (gradient) CKV._ In addition, any gradient proper CKV of the
$M^{n-1}$ provides two proper CKVs for the $M^{n}$. For a four-dimensional
manifold, the following result is shown in [131].

If $M^{n}$, where $n=4$, is a decomposable Riemannian manifold with line
element ($\mu,\nu=1,2,3)$

$ds^{2}=\varepsilon dt^{2}+h_{\mu\nu}\left(x^{\sigma}\right)dx^{\mu}dx^{\nu}$
(6.3)

where $\varepsilon=\pm 1$, then the vector field

$X^{a}\partial_{a}=-\frac{\varepsilon}{p}\dot{\lambda}\left(t\right)\psi\left(x^{\sigma}\right)\partial_{t}+\frac{1}{p}\lambda\left(t\right)\xi^{\mu}\left(x^{\sigma}\right)\partial_{\mu}+L^{\mu}\partial_{\mu}$
(6.4)

is a proper CKV of (6.3), where:
a. $L^{\mu}$ is a non-gradient KV or HV of $M^{n-1}$.
b. $\xi^{\mu}\left(x^{\sigma}\right)$ is a gradient proper CKV of $M^{n-1}$
with conformal factor $\psi\left(x^{\sigma}\right)$, i.e. the Lie derivative
$\mathit{L}_{\boldsymbol{\xi}}h_{\mu\nu}\left(x^{\sigma}\right)=2\psi\left(x^{\sigma}\right)h_{\mu\nu}\left(x^{\sigma}\right)$.
c. The function $\lambda\left(t\right)$ is given by

$\lambda\left(t\right)=\lambda_{1}e^{i\sqrt{\varepsilon
p}t}+\lambda_{2}e^{-i\sqrt{\varepsilon p}t},\enskip\text{for $\varepsilon
p>0$}$ (6.5)

or

$\lambda\left(t\right)=\lambda_{1}e^{\sqrt{-\varepsilon
p}t}+\lambda_{2}e^{-\sqrt{-\varepsilon p}t},\enskip\text{for $\varepsilon
p<0$}$ (6.6)

where $p$ is a non-vanishing constant and $\lambda_{1}$, $\lambda_{2}$ are
independent constants, provided the function $\psi\left(x^{\sigma}\right)$
satisfies the condition

$\psi_{;\mu\nu}=p\psi h_{\mu\nu}.$ (6.7)

Concerning the HV, it has been shown in [131] that when the $M^{n-1}$ space
admits a HV $H^{\mu}\left(x^{\sigma}\right)$ with conformal factor $C$, the
$M^{n}$ admits the HV

$H^{a}\partial_{a}=Ct\partial_{t}+H^{\mu}\partial_{\mu}.$ (6.8)

Finally, concerning the KV fields, it has been shown that the KV fields of
$M^{n}$ are

$\displaystyle K^{a}\partial_{a}$ $\displaystyle=$ $\displaystyle
k_{0}\partial_{t}+\left[k_{1I}h^{\mu\nu}\left(x^{\sigma}\right)K_{\nu}^{I}\left(x^{\sigma}\right)+k_{2I}h^{\mu\nu}\left(x^{\sigma}\right)S_{,\nu}^{I}\left(x^{\sigma}\right)\right]\partial_{\mu}+$
(6.9) $\displaystyle+k_{3I}\left[-\varepsilon
S^{I}\left(x^{\sigma}\right)\partial_{t}+h^{\mu\nu}\left(x^{\sigma}\right)S_{,\nu}^{I}\left(x^{\sigma}\right)\partial_{\mu}\right]$

where $K_{\nu}^{I}\left(x^{\sigma}\right)$ are the non-gradient KVs of
$M^{n-1}$ and $S_{,\nu}^{I}\left(x^{\sigma}\right)$ are the gradient KVs of
$M^{n-1}$. The parameters $k_{0}$, $k_{1I}$, $k_{2I}$ and $k_{3I}$ are
independent constants.

However, another possibility that the space (6.3) admits proper CKVs is when
it is conformally flat. That case was found to be important in the
classification of Bianchi I spacetimes in [109] according to the admitted
CKVs, but it does not provide any result in the case of Bianchi III and
Bianchi V spacetimes. Thus, we omit it from the present discussion.

The concept of conformally related metrics plays a crucial role in the
computation of the CKVs; therefore, we review the basic definitions concerning
these metrics. Two metrics $\widehat{g}_{ab}$ and $g_{ab}$ are said to be
conformally related iff there is a function $N^{2}(x^{r})$ such that
$\widehat{g}_{ab}=N^{2}(x^{r})g_{ab}$. The conformally related metrics share
the same conformal algebra but with different conformal factors. For a given
vector field $\mathbf{X}$, we have the following decompositions/identities
(see sec. 2.2):

$L_{\mathbf{X}}\widehat{g}_{ab}=2\widehat{\psi}(\mathbf{X})\widehat{g}_{ab}+2\widehat{H}_{ab}(\mathbf{X})\enskip\text{and}\enskip
L_{\mathbf{X}}g_{ab}=2\psi(\mathbf{X})g_{ab}+2H_{ab}(\mathbf{X})$

where $H_{ab}(\mathbf{X})$ and $\widehat{H}_{ab}(\mathbf{X})$ are symmetric
traceless tensors. It can be shown that

$\widehat{\psi}(\mathbf{X})=\mathbf{X}(\ln
N)+\psi(\mathbf{X}),\enskip\widehat{H}_{ab}(\mathbf{X})=N^{2}H_{ab}(\mathbf{X})\enskip\text{and}\enskip\widehat{F}_{ab}(\mathbf{X})=N^{2}F_{ab}(\mathbf{X})-2NN_{,[a}X_{b]}$

where
$\widehat{F}_{ab}(\mathbf{X})=\widehat{X}_{[a\hat{;}b]}=\widehat{X}_{[a;b]}$
and $F_{ab}(\mathbf{X})=X_{[a;b]}$. Moreover,

$\widehat{X}_{a\hat{;}b}=\frac{1}{2}L_{\mathbf{X}}\widehat{g}_{ab}+\widehat{F}_{ab}(\mathbf{X})\enskip\text{and}\enskip
X_{a;b}=\frac{1}{2}L_{\mathbf{X}}g_{ab}+F_{ab}(\mathbf{X}).$

A metric $g_{ab}$ is called conformally flat iff it is conformally related to
the flat metric222For more details on the conformal algebra of the flat
metric, see sec. 2.3. $\eta_{ab}$. A metric conformally related to a
conformally flat metric is also conformally flat. It is well-known that all
the 2d spacetimes are conformally flat and admit an infinite number of CKVs,
while only the flat 2d metrics admit SCKVs.

####  6.3 CKVs of Bianchi III spacetimes

Consider the 3d decomposable spacetime of Lorentzian signature

$ds_{(1+2)}^{2}=\Gamma^{2}\left(\tau\right)\left(-d\tau^{2}+dx^{2}\right)+dy^{2}.$
(6.10)

The line element (6.10) for arbitrary $\Gamma\left(\tau\right)$ admits a 2d
conformal Killing algebra consisting by the KVs $\partial_{x}$ and
$\partial_{y}$.

For the conformal spacetime

$d\bar{s}_{\left(1+2\right)}^{2}=B^{2}\left(\tau\right)e^{2x}ds_{(1+2)}^{2}$
(6.11)

the vector field $\partial_{y}$ remains a KV but $\partial_{x}$ now becomes a
proper HV.

Consider now the 4d decomposable spacetime

$ds_{\left(1+3\right)}^{2}=d\bar{s}_{\left(1+2\right)}^{2}+dz^{2}$ (6.12)

which admits a 3d conformal algebra consisting of the KVs
$\partial_{y},\partial_{z}$ and the proper HV $\partial_{x}+z\partial_{z}$.
Then, the conformally related spacetime
$ds_{\left(III\right)}^{2}=A^{2}\left(\tau\right)e^{-2x}ds_{\left(1+3\right)}^{2}$
which can be written equivalently333Here
$\alpha^{2}\left(t\right)=A^{2}\left(t\right)B^{2}\left(t\right)\Gamma^{2}\left(t\right)$,
$\beta^{2}\left(t\right)=A^{2}\left(t\right)B^{2}\left(t\right)$ and
$\gamma^{2}\left(t\right)=A^{2}\left(t\right),$ while $t=\int
a\left(\tau\right)d\tau$.

$ds_{\left(III\right)}^{2}=-dt^{2}+\alpha^{2}\left(t\right)dx^{2}+\beta^{2}\left(t\right)dy^{2}+\gamma^{2}\left(t\right)e^{-2x}dz^{2}$
(6.13)

is a Bianchi III spacetime and the vector fields
$\partial_{y},~{}\partial_{z},~{}\partial_{x}+z\partial_{z}$ form the Killing
algebra of (6.13). Therefore, in order the Bianchi III spacetime (6.13) to
admit a greater conformal algebra, the functions
$\alpha\left(t\right),~{}\beta\left(t\right)$ and $\gamma\left(t\right)$ must
be specified. Recall that when $\alpha\left(t\right)=\gamma\left(t\right)$,
the spacetime (6.13) is locally rotational and admits as extra KV the rotation
in the 2d space $ds^{2}=dx^{2}+e^{-2x}dz^{2}$.

The 3d spacetime (6.10) admits a greater conformal algebra for specific
functions $\Gamma\left(\tau\right)$. From the discussion of sec. 6.2, it
follows that $\Gamma\left(\tau\right)$ must be such that the 2d space

$ds_{\left(2\right)}^{2}=\Gamma^{2}\left(\tau\right)\left(-d\tau^{2}+dx^{2}\right)$
(6.14)

admits proper gradient CKVs or a greater Killing algebra. For 2d spaces, it is
well-known that the admitted KVs can be zero, one or three and, in the latter
case, the space is maximally symmetric. Since (6.14) admits always the KV
$\partial_{x}$, the $\Gamma\left(\tau\right)$ must be specified so that (6.14)
is maximally symmetric. Without loss of generality, we can select either
$\Gamma^{2}\left(\tau\right)=e^{m\tau}$ in which case (6.14) is the flat space
with Ricci Scalar $R_{\left(2\right)}=0$, or
$\Gamma^{2}\left(\tau\right)=\kappa^{-2}\cos^{-2}\left(\tau\right)$ in which
case $R_{\left(2\right)}=2\kappa^{2}$.

Furthermore, because all the 2d spaces admit infinitely many CKVs, the
requirement that at least one of the proper CKVs is to be gradient specifies
the spacetime to be of nonzero constant curvature (i.e. maximally symmetric
space and admits five gradient proper CKVs).

#####  6.3.1 Case $\Gamma^{2}\left(\tau\right)=e^{m\tau}$

In the case of $\Gamma^{2}\left(\tau\right)=e^{m\tau}$, the 3d space

$ds_{(1+2)}^{2}=e^{m\tau}\left(-d\tau^{2}+dx^{2}\right)+dy^{2}.$ (6.15)

is flat (see sec. 2.3) and admits a ten-dimensional conformal algebra. This
algebra consists of the six KVs:

$\mathbf{Y}_{1}=\frac{2}{m}e^{-\frac{m}{2}(\tau-x)}\partial_{\tau}-\frac{2}{m}e^{-\frac{m}{2}(\tau-x)}\partial_{x},\enskip\mathbf{Y}_{2}=-\frac{2}{m}e^{-\frac{m}{2}(\tau+x)}\partial_{\tau}-\frac{2}{m}e^{-\frac{m}{2}(\tau+x)}\partial_{x},$
$\mathbf{Y}_{3}=\partial_{x},\enskip\mathbf{Y}_{4}=\partial_{y},\enskip\mathbf{Y}_{5}=ye^{-\frac{m}{2}(\tau+x)}\partial_{\tau}+ye^{-\frac{m}{2}(\tau+x)}\partial_{x}+\frac{2}{m}e^{\frac{m}{2}(\tau-x)}\partial_{y},$
$\mathbf{Y}_{6}=-ye^{-\frac{m}{2}(\tau-x)}\partial_{\tau}+ye^{-\frac{m}{2}(\tau-x)}\partial_{x}-\frac{2}{m}e^{\frac{m}{2}(\tau+x)}\partial_{y},$

the HV $\mathbf{Y}_{7}=\frac{2}{m}\partial_{\tau}+y\partial_{y}$ with
homothetic factor $\psi_{(1+2)}(\mathbf{Y}_{7})=1$ and the three SCKVs:

$\displaystyle\mathbf{Y}_{8}$ $\displaystyle=$
$\displaystyle\left[\frac{2}{m^{2}}e^{\frac{m}{2}(\tau-x)}+\frac{y^{2}}{2}e^{-\frac{m}{2}(\tau+x)}\right]\partial_{\tau}+\left[-\frac{2}{m^{2}}e^{\frac{m}{2}(\tau-x)}+\frac{y^{2}}{2}e^{-\frac{m}{2}(\tau+x)}\right]\partial_{x}+\frac{2y}{m}e^{\frac{m}{2}(\tau-x)}$
$\displaystyle\mathbf{Y}_{9}$ $\displaystyle=$
$\displaystyle-\left[\frac{2}{m^{2}}e^{\frac{m}{2}(\tau+x)}+\frac{y^{2}}{2}e^{-\frac{m}{2}(\tau-x)}\right]\partial_{\tau}+\left[-\frac{2}{m^{2}}e^{\frac{m}{2}(\tau+x)}+\frac{y^{2}}{2}e^{-\frac{m}{2}(\tau-x)}\right]\partial_{x}-\frac{2y}{m}e^{\frac{m}{2}(\tau+x)}$
$\displaystyle\mathbf{Y}_{10}$ $\displaystyle=$ $\displaystyle
my\partial_{\tau}+\left[\frac{m^{2}y^{2}}{4}+e^{m\tau}\right]\partial_{y}$

with conformal factors
$\psi_{(1+2)}(\mathbf{Y}_{8})=\frac{2}{m}e^{\frac{m}{2}(\tau-x)}$,
$\psi_{(1+2)}(\mathbf{Y}_{9})=-\frac{2}{m}e^{\frac{m}{2}(\tau+x)}$ and
$\psi_{(1+2)}(\mathbf{Y}_{10})=\frac{m^{2}y}{2}$, respectively.

The conformally flat space

$d\bar{s}_{\left(1+2\right)}^{2}=B^{2}\left(\tau\right)e^{2x}\left[e^{m\tau}\left(-d\tau^{2}+dx^{2}\right)+dy^{2}\right]$
(6.16)

admits the same elements of conformal algebra with (6.15), but with different
conformal factors

$\bar{\psi}_{(1+2)}(\mathbf{Y}_{A})=\mathbf{Y}_{A}\left[\ln(Be^{x})\right]+\psi_{(1+2)}(\mathbf{Y}_{A}).$
(6.17)

When we impose the condition (6.7), we find that there does not exist a
function $B\left(\tau\right)$ such that the factors
$\bar{\psi}_{(1+2)}(\mathbf{Y}_{A})$ to satisfy (6.7). On the other hand, we
observe that for

$B\left(\tau\right)=e^{\mu\tau},\enskip\mu=\frac{m(\lambda-1)}{2}$ (6.18)

it follows $\bar{\psi}_{(1+2)}(\mathbf{Y}_{7})=\lambda=const$, which means
that $Y_{7}$ is reduced to a HV for (6.16). At this point, it is important to
mention that $\bar{\psi}_{1+2}\left(\mathbf{Y}_{3}\right)=1$; however, as
expected, there is only one proper HV and not two. We assume $\mathbf{Y}_{7}$
to be the proper HV and $\mathbf{Y}_{3}-\frac{1}{\lambda}\mathbf{Y}_{7}$ to be
a KV.

For the 4d decomposable spacetime

$ds_{(1+3)}^{2}=e^{2x}e^{2\mu\tau}\left[e^{m\tau}\left(-d\tau^{2}+dx^{2}\right)+dy^{2}\right]+dz^{2}$
(6.19)

from $\mathbf{Y}_{7}$, we find the proper HV

$\mathbf{L}_{1}\equiv\mathbf{Y}_{7}+\lambda
z\partial_{z}=\frac{2}{m}\partial_{\tau}+y\partial_{y}+\lambda z\partial_{z}.$
(6.20)

We conclude that the Bianchi III spacetime

$ds_{(III)}^{2}=e^{m\lambda\tau}A^{2}\left(\tau\right)\left(-d\tau^{2}+dx^{2}+e^{-m\tau}dy^{2}+e^{-m\lambda\tau}e^{-2x}dz^{2}\right)$
(6.21)

admits the proper CKV $\mathbf{L}_{1}$ with conformal factor
$\psi_{(III)}(\mathbf{L}_{1})=\frac{2}{m}\frac{A_{,\tau}}{A}+\lambda$, which
reduces to a HV when $A\left(\tau\right)$ is an exponential. In the last case,
the line element is

$ds_{(III)}^{2}=-e^{m\kappa\tau}d\tau^{2}+e^{m\kappa\tau}dx^{2}+e^{m(\kappa-1)\tau}dy^{2}+e^{m(\kappa-\lambda)\tau}e^{-2x}dz^{2}$
(6.22)

or, equivalently,

$ds_{(III)}^{2}=-dt^{2}+\frac{m^{2}\kappa^{2}t^{2}}{4}dx^{2}+\left(\frac{m^{2}\kappa^{2}t^{2}}{4}\right)^{\frac{\kappa-1}{\kappa}}dy^{2}+\left(\frac{m^{2}\kappa^{2}t^{2}}{4}\right)^{\frac{\kappa-\lambda}{\kappa}}e^{-2x}dz^{2}$
(6.23)

and we write $\mathbf{L}_{1}=\kappa t\partial_{t}+y\partial_{y}+\lambda
z\partial_{z}$ with $\psi_{(III)}(\mathbf{L}_{1})=const\equiv\kappa\neq 0$.
Recall that $dt=e^{\frac{m\kappa}{2}\tau}d\tau$.

Performing the same analysis for the second case of
$\Gamma^{2}\left(\tau\right)=\kappa^{-2}\cos^{-2}\left(\tau\right)$, we find
that the resulting Bianchi III spacetime does not admit any proper CKV or a
proper HV; hence, we omit the presentation of this analysis.

We summarize our results in the following Proposition.

###### Proposition 6.3.1

The only Bianchi III spacetime which admits a proper CKV is

$ds^{2}=A^{2}\left(\tau\right)\left[e^{m\lambda\tau}\left(-d\tau^{2}+dx^{2}\right)+e^{m\left(\lambda-1\right)\tau}dy^{2}+e^{-2x}dz^{2}\right].$
(6.24)

The CKV is $\mathbf{L}_{1}=\frac{2}{m}\partial_{\tau}+y\partial_{y}+\lambda
z\partial_{z}$ and it has conformal factor
$\psi_{(III)}(\mathbf{L}_{1})=\frac{2}{m}\frac{A_{,\tau}}{A}+\lambda$, where
$A\left(\tau\right)$ is an arbitrary function.

####  6.4 Bianchi V spacetimes which admit a CKV

For the computation of the CKVs for the Bianchi V spacetime, we apply the same
procedure with sec. 6.3, but for this case, we start from the 2d spacetime

$ds_{\left(2\right)}^{2}=\Gamma^{2}\left(\tau\right)e^{-2x}\left(-d\tau^{2}+dx^{2}\right).$
(6.25)

The latter space is maximally symmetric only for
$\Gamma^{2}\left(\tau\right)=e^{\gamma\tau}$, where the Ricci Scalar is
calculated to be $R_{\left(2\right)}=0$. It is important to mention that there
does not exist a function $\Gamma\left(\tau\right)$ such that the space (6.25)
is of constant curvature.

We omit the intermediary calculations and we summarize the results in the
following Proposition.

###### Proposition 6.4.1

The Bianchi V spacetime

$ds^{2}=A^{2}\left(\tau\right)\left[\Gamma^{2}\left(\tau\right)\left(-d\tau^{2}+dx^{2}\right)+e^{2x}\left(B^{2}\left(\tau\right)dy^{2}+dz^{2}\right)\right]$
(6.26)

admits the unique proper CKV
$\mathbf{L}_{1}=\frac{2}{m}\partial_{\tau}+y\partial_{y}+\lambda
z\partial_{z}$ with
$\psi_{(V)}(\mathbf{L}_{1})=\frac{2}{m}\frac{A_{,\tau}}{A}+\lambda$ only when
$\Gamma^{2}\left(\tau\right)=e^{m\lambda\tau}$ and
$B^{2}\left(\tau\right)=e^{m(\lambda-1)\tau}$. For
$A^{2}\left(\tau\right)=e^{m(\kappa-\lambda)\tau}$, the CKV reduces to a HV
with conformal factor $\psi_{(V)}(\mathbf{L}_{1})=const=\kappa\neq 0$.

####  6.5 Applications

#####  6.5.1 Bianchi III cosmological fluid

In this section, we study some of the kinematic and the dynamic properties of
spacetime given by equation (6.21) for the comoving observers
$u^{a}=\frac{e^{-\frac{m\lambda}{2}\tau}}{A\left(\tau\right)}\delta_{\tau}^{a}$,
where $u^{a}u_{a}=-1$. As it is well-known (see e.g. [92] and chapter 5), the
four-velocity of a class of observers introduces the $1+3$ decomposition of
tensor fields in spacetime. The decomposition of $u_{a;b}$ gives the kinematic
quantities $\theta,~{}\sigma^{2},~{}\omega^{2}$ and $\alpha^{a}$ defined by
the identity

$u_{a;b}=-\alpha_{a}u_{b}+\omega_{ab}+\sigma_{ab}+\frac{1}{3}\theta h_{ab}$
(6.27)

where $\alpha^{a}=\dot{u}^{a}=u^{a}{}_{;b}u^{b}$,
$\omega_{ab}=h_{a}^{c}h_{b}^{d}u_{[c;d]}$,
$\sigma_{ab}=\left(h_{a}^{c}h_{b}^{d}-\frac{1}{3}h^{cd}h_{ab}\right)u_{(c;d)}$,
$\theta=h^{ab}u_{a;b}=u^{a}{}_{;a}$,
$\sigma^{2}\equiv\frac{1}{2}\sigma_{ab}\sigma^{ab}$ and
$\omega^{2}\equiv\frac{1}{2}\omega_{ab}\omega^{ab}.$ In this decomposition,
$\alpha^{a}$ is the four-acceleration of the observers $u^{a}$, and the
quantities $\omega_{ab},\sigma_{ab},\theta$ concern the variation of the
projected ($\bot\delta x^{a}u_{a}=0$) connecting vector $\bot\delta x^{a}$
along the congruence (i.e. the integral lines) defined by the vector field
$u^{a}$. The antisymmetric tensor $\omega_{ab}$ measures the relative
rotation; the tensor $\sigma_{ab}$, the anisotropic expansion; and the scalar
$\theta$, the isotropic expansion of $\bot\delta x^{a}$. The dynamic variables
of the spacetime are defined by the 1+3 decomposition of the Einstein tensor
$G_{ab}$ as follows [92]:

$G_{ab}=\rho u_{a}u_{b}+2q_{(a}u_{b)}+ph_{ab}+\pi_{ab}$ (6.28)

where $\rho=G_{ab}u^{a}u^{b}$ is the energy-mass density of the fluid,
$p=\frac{1}{3}h^{ab}G_{ab}$ is the isotropic pressure,
$q^{a}=-h^{ac}G_{cd}u^{d}$ is the heat flux tensor and
$\pi_{ab}=\left(h_{a}^{c}h_{b}^{d}-\frac{1}{3}h^{cd}h_{ab}\right)G_{cd}$ is
the traceless anisotropic tensor (measures the anisotropy of the fluid).

Applying the above for the comoving observers in Bianchi III spacetime (6.21),
we compute that the kinematic quantities are $\omega^{2}=0$, $\alpha^{a}=0$,

$\theta=\frac{e^{-\frac{m\lambda}{2}\tau}}{A}\left[3\frac{d(\ln
A)}{d\tau}+\frac{m(2\lambda-1)}{2}\right]$ (6.29)

and

$\sigma^{2}=\frac{m^{2}(\lambda^{2}-\lambda+1)}{12}\frac{e^{-m\lambda\tau}}{A^{2}}.$
(6.30)

Similarly for the dynamic quantities, we find that the (nonzero) components
for the cosmological fluid defined by the Bianchi III spacetime (6.21) are

$\rho=\frac{e^{-m\lambda\tau}}{4A^{2}}\left[4\frac{d\left(\ln
A\right)}{d\tau}\left(3\frac{d\left(\ln
A\right)}{d\tau}+m\left(2\lambda-1\right)\right)+m^{2}\lambda\left(\lambda-1\right)-4\right]$
(6.31)
$p=\frac{e^{-m\lambda\tau}}{A^{2}}\left[-\frac{2}{A}\frac{d^{2}A}{d\tau^{2}}+\frac{d\left(\ln
A\right)}{d\tau}\left(\frac{d\left(\ln
A\right)}{d\tau}+\frac{m}{3}(2-\lambda)\right)-\frac{m^{2}}{12}(\lambda-1)(\lambda-2)+\frac{1}{3}\right]$
(6.32)
$q^{a}=\left(0,\frac{m\lambda}{2}\frac{e^{-\frac{3m\lambda}{2}\tau}}{A^{3}},0,0\right)$
(6.33) $\pi_{xx}=\frac{m(\lambda+1)}{3}\frac{d\left(\ln
A\right)}{d\tau}+\frac{m^{2}(\lambda^{2}-1)}{12}-\frac{1}{3}$ (6.34)
$\pi_{yy}=e^{-m\tau}\left[\frac{m(\lambda-2)}{3}\frac{d\left(\ln
A\right)}{d\tau}+\frac{m^{2}}{12}(\lambda-1)(\lambda-2)+\frac{2}{3}\right]$
(6.35)

and

$\pi_{zz}=-e^{-m\lambda\tau-2x}\left[\frac{m(2\lambda-1)}{3}\frac{d\left(\ln
A\right)}{d\tau}+\frac{m^{2}}{12}(\lambda-1)(2\lambda-1)+\frac{1}{3}\right].$
(6.36)

In the case of $A^{2}(\tau)=e^{m(\kappa-\lambda)\tau}$, where the CKV
$\mathbf{L}_{1}$ becomes a HV, the above nonzero quantities are simplified as
follows:

$\theta=\frac{m(3\kappa-\lambda-1)}{2}e^{-\frac{m\kappa}{2}\tau}$ (6.37)
$\sigma^{2}=\frac{m^{2}(\lambda^{2}-\lambda+1)}{12}e^{-m\kappa\tau}$ (6.38)
$\rho=\rho_{0}\left(m,\kappa,\lambda\right)e^{-m\kappa\tau}~{},~{}p=p_{0}\left(m,\kappa,\lambda\right)e^{-m\kappa\tau}$
(6.39)
$q^{a}=\left(0,\frac{m\lambda}{2}e^{-\frac{3m\kappa}{2}\tau},0,0\right)$
(6.40)

and

$\pi_{xx}=\pi_{xx0}\left(m,\kappa,\lambda\right)~{},~{}\pi_{yy}=\pi_{yy0}\left(m,\kappa,\lambda\right)e^{-m\tau}~{},~{}\pi_{zz}=\pi_{zz0}\left(m,\kappa,\lambda\right)e^{-m\lambda\tau-2x}.$
(6.41)

From the latter expressions, we infer that for large $\tau$ and $m\kappa>0$,
all the kinematical quantities, the mass density, the isotropic pressure and
the heat flux vector vanish. If, in addition,
$\pi_{xx0}\left(m,\kappa,\lambda\right)=0$, $m>0$ and $\lambda>0$, then, for
large $\tau$, the fluid source vanishes and the solution describes an
isotropic empty spacetime.

#####  6.5.2 Bianchi V cosmological fluid

We consider the extended Bianchi V spacetime of Proposition 6.4.1

$ds_{(V)}^{2}=-A^{2}(\tau)e^{m\lambda\tau}d\tau^{2}+A^{2}(\tau)e^{m\lambda\tau}dx^{2}+A^{2}(\tau)e^{m(\lambda-1)\tau}e^{2x}dy^{2}+A^{2}(\tau)e^{2x}dz^{2}$
(6.42)

and repeat the calculations of the previous section for the comoving
observers. We find that the kinematic quantities are exactly the same with
those of the Bianchi III spacetime, while the (nonzero) dynamic variables of
the cosmological fluid are

$\rho=\frac{e^{-m\lambda\tau}}{4A^{2}}\left[4\frac{d\left(\ln
A\right)}{d\tau}\left(3\frac{d\left(\ln
A\right)}{d\tau}+m\left(2\lambda-1\right)\right)+m^{2}\lambda\left(\lambda-1\right)-12\right]$
(6.43)
$p=\frac{e^{-m\lambda\tau}}{A^{2}}\left[-\frac{2}{A}\frac{d^{2}A}{d\tau^{2}}+\frac{d\left(\ln
A\right)}{d\tau}\left(\frac{d\left(\ln
A\right)}{d\tau}+\frac{m}{3}(2-\lambda)\right)-\frac{m^{2}}{12}(\lambda-1)(\lambda-2)+1\right]$
(6.44)
$q^{a}=\left(0,-\frac{m(\lambda+1)}{2}\frac{e^{-\frac{3m\lambda}{2}\tau}}{A^{3}},0,0\right)$
(6.45) $\pi_{xx}=\frac{m(\lambda+1)}{3}\frac{d\left(\ln
A\right)}{d\tau}+\frac{m^{2}(\lambda^{2}-1)}{12}$ (6.46)
$\pi_{yy}=e^{-m\tau+2x}\left[\frac{m(\lambda-2)}{3}\frac{d\left(\ln
A\right)}{d\tau}+\frac{m^{2}}{12}(\lambda-1)(\lambda-2)\right]$ (6.47)

and

$\pi_{zz}=-e^{-m\lambda\tau+2x}\left[\frac{m(2\lambda-1)}{3}\frac{d\left(\ln
A\right)}{d\tau}+\frac{m^{2}}{12}(\lambda-1)(2\lambda-1)\right].$ (6.48)

In the case $\mathbf{L}_{1}$ is a HV, we deduce the same conclusions with the
Bianchi III case of sec. 6.5.1.

#####  6.5.3 Lie point symmetries of the wave equation

Collineations of spacetimes can be used to construct symmetries and
conservation laws for some differential equations defined in curved
spacetimes. In [139], it has been shown that there exists a unique connection
between the point Noether symmetries for the geodesic Lagrangian of a given
Riemannian space and the elements of the admitted homothetic algebra. Similar
results have been proved for other PDEs of special interest [11, 140].

In this section, we consider the wave equation

$\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{-g}g^{\mu\nu}\frac{\partial}{\partial
x^{\nu}}\right)u\left(x^{\lambda}\right)=0$ (6.49)

in the Bianchi III spacetime (6.24) and in the Bianchi V spacetime (6.26), and
determine its Lie point symmetries. By following the generic results of [140],
we find that the admitted Lie point symmetries of the wave equation in the
Bianchi III spacetime (6.24) are the three KVs, the vector field
$Y_{u}=u\partial_{u}$, and the infinitely many vectors
$Y_{\infty}=b\left(x^{\mu}\right)\partial_{u}$, where $b\left(x^{\mu}\right)$
is a solution of the original equation (6.49). The latter symmetry vector
fields exist because equation (6.49) is a linear PDE.

For a higher-dimensional conformal algebra, equation (6.49) admits extra Lie
point symmetries. Indeed, from our analysis and for the case where the Bianchi
III and Bianchi V spacetimes admit a proper HV, the wave equation becomes,
respectively,

$\left(-u_{tt}+u_{xx}+u_{yy}+e^{m\lambda
t+2x}u_{zz}\right)+\frac{m}{2}\left(\lambda-2\kappa+1\right)u_{t}-u_{x}=0$
(6.50)

and

$\left(-u_{tt}+u_{xx}+e^{mt-2x}u_{yy}+e^{m\lambda
t-2x}u_{zz}\right)+\frac{m}{2}\left(\lambda+2\kappa-1\right)u_{t}+4u_{x}=0.$
(6.51)

Then, we find that equation (6.50) admits the generic Lie point symmetry
vector

$Y_{III}=\left(a_{1}\frac{2}{m}\right)\partial_{t}+a_{2}\partial_{x}+\left(a_{1}y+a_{3}\right)\partial_{y}+\left(a_{1}\lambda
z+a_{2}z+a_{4}\right)\partial_{z}+\left[a_{u}u+a_{\infty}b\left(t,x,y,z\right)\right]\partial_{u}$
(6.52)

while equation (6.51) is invariant under the one-parameter point
transformation with generator

$Y_{V}=\left(a_{1}\frac{2}{m}\right)\partial_{t}+a_{2}\partial_{x}+\left(a_{1}y-a_{1}y+a_{3}\right)\partial_{y}+\left(a_{1}\lambda
z-a_{2}z+a_{4}\right)\partial_{z}+\left[a_{u}u+a_{\infty}b\left(t,x,y,z\right)\right]\partial_{u}.$
(6.53)

The latter symmetry vectors can be applied to construct conservation laws or
similarity solutions for the wave equation. However, such an analysis is
beyond the scope of this work.

####  6.6 Summary of results

The Bianchi I spacetimes:

  1. 1.

The spacetime

$ds^{2}_{(I)}=-C^{2}e^{m\lambda\tau}d\tau^{2}+C^{2}e^{m\lambda\tau}dx^{2}+C^{2}e^{m(\lambda-1)\tau}dy^{2}+C^{2}dz^{2}$

admits the proper CKV
$\mathbf{K}_{2}=\frac{2}{m}\partial_{\tau}+y\partial_{y}+\lambda
z\partial_{z}$ with
$\psi_{(I)}(\mathbf{K}_{2})=\frac{2}{m}\frac{C_{,\tau}}{C}+\lambda$. When
$\mathbf{K}_{2}$ is a HV, then

$\displaystyle ds^{2}_{(I)}$
$\displaystyle=-e^{m\kappa\tau}d\tau^{2}+e^{m\kappa\tau}dx^{2}+e^{m(\kappa-1)\tau}dy^{2}+e^{m(\kappa-\lambda)\tau}dz^{2}$
$\displaystyle=-dt^{2}+\frac{m^{2}\kappa^{2}t^{2}}{4}dx^{2}+\left(\frac{m^{2}\kappa^{2}t^{2}}{4}\right)^{\frac{\kappa-1}{\kappa}}dy^{2}+\left(\frac{m^{2}\kappa^{2}t^{2}}{4}\right)^{\frac{\kappa-\lambda}{\kappa}}dz^{2}$

and $\mathbf{K}_{2}=\kappa t\partial_{t}+y\partial_{y}+\lambda z\partial_{z}$
with $\psi_{(I)}(\mathbf{K}_{2})=const\equiv\kappa\neq 0$.

  2. 2.

The spacetime

$\displaystyle ds^{2}_{(I)}$
$\displaystyle=-C^{2}b^{2}\tau^{-2}d\tau^{2}+C^{2}b^{2}\tau^{-2}dx^{2}+C^{2}b^{2}\tau^{-2c}dy^{2}+C^{2}dz^{2}$
$\displaystyle=-C^{2}dt^{2}+C^{2}b^{2}e^{-2t/b}dx^{2}+C^{2}b^{2}e^{-2ct/b}dy^{2}+C^{2}dz^{2}$

admits the proper CKV
$\mathbf{K}=\tau\partial_{\tau}+x\partial_{x}+cy\partial_{y}$ $=$
$b\partial_{t}+x\partial_{x}+cy\partial_{y}$ with
$\psi_{(I)}(\mathbf{K})=\tau\frac{C_{,\tau}}{C}=b\frac{\dot{C}}{C}$. When
$\mathbf{K}$ is a HV, then

$\displaystyle ds^{2}_{(I)}$
$\displaystyle=-e^{2\psi_{0}t}dt^{2}+e^{2\psi_{0}t}b^{2}e^{-2t/b}dx^{2}+e^{2\psi_{0}t}b^{2}e^{-2ct/b}dy^{2}+e^{2\psi_{0}t}dz^{2}$
$\displaystyle=-d\bar{t}^{2}+b^{2}(\psi_{0}\bar{t})^{\frac{2(\psi_{0}b-1)}{\psi_{0}b}}dx^{2}+b^{2}(\psi_{0}\bar{t})^{\frac{2(\psi_{0}b-c)}{\psi_{0}b}}dy^{2}+(\psi_{0}\bar{t})^{2}dz^{2}.$

  3. 3.

The spacetime

$\displaystyle ds^{2}_{(I)}$
$\displaystyle=-C^{2}b_{2}^{2}\tau^{2(b_{1}-1)}d\tau^{2}+C^{2}b_{2}^{2}\tau^{2(b_{1}-1)}dx^{2}+C^{2}b_{2}^{2}\tau^{2(b_{1}-c)}dy^{2}+C^{2}dz^{2}$
$\displaystyle=-C^{2}dt^{2}+C^{2}b_{1}^{2}\left(\frac{b_{1}}{b_{2}}\right)^{-\frac{2}{b_{1}}}t^{\frac{2(b_{1}-1)}{b_{1}}}dx^{2}+C^{2}b_{1}^{2}\left(\frac{b_{1}}{b_{2}}\right)^{-\frac{2c}{b_{1}}}t^{\frac{2(b_{1}-c)}{b_{1}}}dy^{2}+C^{2}dz^{2}$

admits the proper CKV
$\mathbf{K}_{1}=\tau\partial_{\tau}+x\partial_{x}+cy\partial_{y}+b_{1}z\partial_{z}=b_{1}t\partial_{t}+x\partial_{x}+cy\partial_{y}+b_{1}z\partial_{z}$
with
$\psi_{(I)}(\mathbf{K}_{1})=\tau\frac{C_{,\tau}}{C}+b_{1}=b_{1}t\frac{\dot{C}}{C}+b_{1}$.
When $\mathbf{K}_{1}$ is a HV, then $C=t^{\frac{\psi_{0}-b_{1}}{b_{1}}}$.

  4. 4.

The spacetimes

$ds^{2}_{(RT)}=-dt^{2}+\sin^{2}\left(\frac{t}{a}\right)dx^{2}+\cos^{2}\left(\frac{t}{a}\right)dy^{2}+dz^{2}$

and

$ds^{2}_{(ART)}=-dt^{2}+\sinh^{2}\left(\frac{t}{a}\right)dx^{2}+\cosh^{2}\left(\frac{t}{a}\right)dy^{2}+dz^{2}$

admit a 15-dimensional conformal algebra with a 7-dimensional Killing
subalgebra (see appendix 2 in [109]); its 3-dimensional space is of constant
curvature.

The Bianchi III spacetimes:

  1. 1.

The spacetime

$ds^{2}_{(III)}=-C^{2}(\tau)e^{m\lambda\tau}d\tau^{2}+C^{2}(\tau)e^{m\lambda\tau}dx^{2}+C^{2}(\tau)e^{m(\lambda-1)\tau}dy^{2}+C^{2}(\tau)e^{-2x}dz^{2}$

admits the proper CKV
$\mathbf{L}_{1}=\frac{2}{m}\partial_{\tau}+y\partial_{y}+\lambda
z\partial_{z}$ with
$\psi_{(III)}(\mathbf{L}_{1})=\frac{2}{m}\frac{C_{,\tau}}{C}+\lambda$. When
$\mathbf{L}_{1}$ is a HV, then

$\displaystyle ds^{2}_{(III)}$
$\displaystyle=-e^{m\kappa\tau}d\tau^{2}+e^{m\kappa\tau}dx^{2}+e^{m(\kappa-1)\tau}dy^{2}+e^{m(\kappa-\lambda)\tau}e^{-2x}dz^{2}$
$\displaystyle=-dt^{2}+\frac{m^{2}\kappa^{2}t^{2}}{4}dx^{2}+\left(\frac{m^{2}\kappa^{2}t^{2}}{4}\right)^{\frac{\kappa-1}{\kappa}}dy^{2}+\left(\frac{m^{2}\kappa^{2}t^{2}}{4}\right)^{\frac{\kappa-\lambda}{\kappa}}e^{-2x}dz^{2}$

and $\mathbf{L}_{1}=\kappa t\partial_{t}+y\partial_{y}+\lambda z\partial_{z}$
with $\psi_{(III)}(\mathbf{L}_{1})=const\equiv\kappa\neq 0$.

  2. 2.

The spacetimes

$ds^{2}_{(III)}=-B^{2}e^{m\tau}d\tau^{2}+B^{2}e^{m\tau}dx^{2}+B^{2}dy^{2}+C^{2}e^{-2x}dz^{2}$

and

$ds^{2}_{(III)}=-\frac{16\alpha^{2}C^{2}e^{\tau}}{(e^{\tau}-8c)^{2}}d\tau^{2}+\frac{16\alpha^{2}C^{2}e^{\tau}}{(e^{\tau}-8c)^{2}}dx^{2}+\alpha^{2}C^{2}dy^{2}+C^{2}e^{-2x}dz^{2}$

admit the KV $\mathbf{L}=\lambda\partial_{x}+\lambda z\partial_{z}$.

The Bianchi V spacetimes:

  1. 1.

The spacetime

$ds^{2}_{(V)}=-C^{2}e^{\lambda\tau}d\tau^{2}+C^{2}e^{\lambda\tau}dx^{2}+C^{2}e^{(\lambda-1)\tau}e^{2x}dy^{2}+C^{2}e^{2x}dz^{2}$

admits the proper CKV $\mathbf{L}_{2}=2\partial_{\tau}+y\partial_{y}+\lambda
z\partial_{z}$ with $\psi_{(V)}(\mathbf{L}_{2})=2\frac{C_{\tau}}{C}+\lambda$.
When $\mathbf{L}_{2}$ is a HV, then

$\displaystyle ds^{2}_{(V)}$
$\displaystyle=-e^{\kappa\tau}d\tau^{2}+e^{\kappa\tau}dx^{2}+e^{(\kappa-1)\tau}e^{2x}dy^{2}+e^{(\kappa-\lambda)\tau}e^{2x}dz^{2}$
$\displaystyle=-dt^{2}+\frac{\kappa^{2}t^{2}}{4}dx^{2}+\left(\frac{\kappa^{2}t^{2}}{4}\right)^{\frac{\kappa-1}{\kappa}}e^{2x}dy^{2}+\left(\frac{\kappa^{2}t^{2}}{4}\right)^{\frac{\kappa-\lambda}{\kappa}}e^{2x}dz^{2}$

and $\mathbf{L}_{2}=\kappa t\partial_{t}+y\partial_{y}+\lambda z\partial_{z}$
with $\psi_{(V)}(\mathbf{L}_{2})=const\equiv\kappa\neq 0$.

  2. 2.

Any Bianchi V spacetime

$ds^{2}_{(V)}=-dt^{2}+A^{2}(t)dx^{2}+B^{2}(t)e^{2x}dy^{2}+C^{2}(t)e^{2x}dz^{2}$

admits the KV $\mathbf{M}_{2}=\partial_{x}-y\partial_{y}-z\partial_{z}$.

####  6.7 Conclusions

In this chapter, we have shown that there is only one type of Bianchi III and
Bianchi V spacetime given, respectively, in (6.21) and (6.26) which admit a
single proper CKV. Furthermore, two more spacetimes are found which admit a
HV. In order to arrive at this result, we applied an algorithm which relates
the CKVs of decomposable spacetimes with the collineations of their non-
decomposable subspaces. The kinematics of the fluid of the comoving observers
in all these four spacetimes is not accelerating and rotating and has only
expansion and shear; a result compatible with the anisotropy of the Bianchi
spacetimes. Concerning the dynamics, it has been shown that the fluid of these
observers is heat conducting and anisotropic, that is, it is a general fluid.
Finally, we have used the CKVs we found in each case in order to determine the
generators of the Lie point symmetries of the wave equation in the Bianchi III
spacetime (6.21) and in the Bianchi V spacetime (6.26).

##  Part III Integrability of autonomous dynamical systems

###  Chapter 7 Quadratic first integrals of autonomous conservative dynamical
systems

####  7.1 Introduction

As we have seen in chapter 3, FIs are used to reduce the order of the
dynamical equations. The standard method to determine the FIs of a Lagrangian
dynamical system is to use a special class of Lie symmetries, the Noether
symmetries. We recall that a Noether symmetry is a Lie symmetry which in
addition satisfies the Noether condition [9, 141, 142, 143] (see also sec.
1.4)

$\mathbf{X}^{[1]}L+\frac{d\xi}{dt}L=\frac{df}{dt}.$ (7.1)

According to Noether’s theorem 1.2.1, to every Noether symmetry corresponds
the Noether FI

$I=\xi\left(\dot{q}^{a}\frac{\partial
L}{\partial\dot{q}^{a}}-L\right)-\eta^{a}\frac{\partial
L}{\partial\dot{q}^{a}}+f$ (7.2)

which can be easily determined if one knows the generator
$\mathbf{X}=\xi(t,q,\dot{q},...)\partial_{t}+\eta^{a}(t,q,\dot{q},....)\partial_{q^{a}}$
of the Noether symmetry.

In this chapter, we restrict to autonomous conservative dynamical systems and
we determine their quadratic FIs (QFIs) by following a different approach (see
e.g. [8, 38, 143, 144]). In this latter approach, one assumes the QFI to be of
the general form111 The case of linear FIs (LFIs) also included for
$K_{ab}=0$. :

$I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ (7.3)

where the coefficients $K_{ab},K_{a},K$ are symmetric tensors depending on the
coordinates $t,q^{a}$ and imposes directly the condition $\frac{dI}{dt}=0$
along the dynamical equations. This condition leads to a system of PDEs
involving the unknown quantities $K_{ab},K_{a},K$ whose solution provides the
QFIs (7.3). In all occasions considered so far, the system of these conditions
has been solved for specific cases only. The aim of the present chapter222A
recent preliminary work along this line is presented in [145]. is double: a.
To give the general solution of the system of PDEs resulting from the
condition $\frac{dI}{dt}=0$; and b. To geometrize the answer to the maximum
possible degree.

####  7.2 The conditions for a QFI

#####  7.2.1 The case of a general dynamical system

In this section, we consider an $n$-dimensional dynamical system defined by
the equations of motion

$\ddot{q}^{a}=-\Gamma^{a}_{bc}(q)\dot{q}^{b}\dot{q}^{c}-V^{,a}(q)+Q^{a}\left(t,q,\dot{q}\right)$
(7.4)

where $Q^{a}$ are the non-conservative generalized forces, $\Gamma^{a}_{bc}$
are the Riemannian connection coefficients defined from the kinetic metric
$\gamma_{ab}(q)$ (kinetic energy) of the system and $-V^{,a}$ are the
conservative generalized forces.

We consider, next, a function $I(t,q^{a},\dot{q}^{a})$, which is linear and
quadratic in the velocities, with coefficients which depend only on the
coordinates $t,q^{a}$, that is, $I$ is of the form:

$I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ (7.5)

where $K_{ab}$ is a symmetric tensor, $K_{a}$ is a vector and $K$ is an
invariant.

We demand that $I$ is a FI of (7.4). This requirement leads to the condition
$\frac{dI}{dt}=0$ which gives a system of PDEs for the coefficients $K_{ab},$
$K_{a}$ and $K$. Using the dynamical equations (7.4) to replace $\ddot{q}^{a}$
whenever it appears, we find

$\displaystyle\frac{dI}{dt}$
$\displaystyle=K_{(ab;c)}\dot{q}^{a}\dot{q}^{b}\dot{q}^{c}+\left(K_{ab,t}+K_{a;b}\right)\dot{q}^{a}\dot{q}^{b}+2K_{ab}\dot{q}^{(b}(Q^{a)}-V^{,a)})+\left(K_{a,t}+K_{,a}\right)\dot{q}^{a}+$
$\displaystyle\quad+K_{a}(Q^{a}-V^{,a})+K_{,t}.$ (7.6)

In order to get a working environment, we restrict our considerations to
linear generalized forces, that is, we consider the case
$Q^{a}=A_{b}^{a}(q)\dot{q}^{b}$. Then, the general result (7.6) becomes

$\displaystyle 0$
$\displaystyle=K_{(ab;c)}\dot{q}^{a}\dot{q}^{b}\dot{q}^{c}+\left(K_{ab,t}+K_{a;b}+2K_{c(b}A_{a)}^{c}\right)\dot{q}^{a}\dot{q}^{b}+\left(K_{a,t}+K_{,a}-2K_{ab}V^{,b}+\right.$
$\displaystyle\quad\left.+K_{b}A_{a}^{b}\right)\dot{q}^{a}+K_{,t}-K_{a}V^{,a}$

from which follows the system of PDEs333A subcase of these equations, for
$K_{a}=0$ and $A^{a}_{b}=0$, has been found before by e.g. Kalotas (see eqs.
(12a) - (12d) in [143]) who considered their solution in certain special
cases.:

$\displaystyle K_{(ab;c)}$ $\displaystyle=$ $\displaystyle 0$ (7.7)
$\displaystyle K_{ab,t}+K_{(a;b)}+2K_{c(b}A_{a)}^{c}$ $\displaystyle=$
$\displaystyle 0$ (7.8)
$\displaystyle-2K_{ab}V^{,b}+K_{a,t}+K_{,a}+K_{b}A_{a}^{b}$ $\displaystyle=$
$\displaystyle 0$ (7.9) $\displaystyle K_{,t}-K_{a}V^{,a}$ $\displaystyle=$
$\displaystyle 0.$ (7.10)

Condition $K_{(ab;c)}=0$ implies that $K_{ab}$ is a KT of order two (possibly
zero) of the kinetic metric $\gamma_{ab}$.

Because $\gamma_{ab}$ is autonomous, the condition $K_{(ab;c)}=0$ is satisfied
if the KT $K_{ab}$ is of the form

$K_{ab}(t,q)=g(t)C_{ab}(q)$ (7.11)

where $g(t)$ is an arbitrary analytic function and $C_{ab}(q)$ is a KT of
order two of the metric $\gamma_{ab}$. This choice of $K_{ab}$ and equation
(7.8) indicate that we set

$K_{a}(t,q)=f(t)L_{a}(q)+B_{a}(q)$ (7.12)

where $f(t)$ is an arbitrary analytic function and $L_{a}(q),B_{a}(q)$ are
arbitrary vectors.

Replacing the choices (7.11) and (7.12) in the system of equations (7.7) -
(7.10), we find the following system of PDEs444Equation (7.7) is satisfied
identically, because the quantities $C_{ab}(q)$ are assumed to second order
KTs.:

$\displaystyle g_{,t}C_{ab}+f(t)L_{(a;b)}+B_{(a;b)}+2g(t)C_{c(b}A_{a)}^{c}$
$\displaystyle=$ $\displaystyle 0$ (7.13)
$\displaystyle-2g(t)C_{ab}V^{,b}+f_{,t}L_{a}+K_{,a}+(fL_{b}+B_{b})A_{a}^{b}$
$\displaystyle=$ $\displaystyle 0$ (7.14) $\displaystyle
K_{,t}-(fL_{a}+B_{a})V^{,a}$ $\displaystyle=$ $\displaystyle 0.$ (7.15)

Conditions (7.13) - (7.15) must be supplemented with the integrability
conditions $K_{,at}=K_{,ta}$ and $K_{,[ab]}=0$ for the scalar function $K$.
The integrability condition $K_{,at}=K_{,ta}$ gives –if we make use of (7.14)
and (7.15)– the equation

$f_{,tt}L_{a}+f_{,t}L_{b}A_{a}^{b}+f\left(L_{b}V^{;b}\right)_{;a}+\left(B_{b}V^{;b}\right)_{;a}-2g_{,t}C_{ab}V^{,b}=0.$
(7.16)

Condition $K_{,[ab]}=0$ gives the equation

$2g\left(C_{[a\left|c\right|}V^{,c}\right)_{;b]}-f_{,t}L_{\left[a;b\right]}-(fL_{c;[b}+B_{c;[b})A_{a]}^{c}-(fL_{c}+B_{c})A_{[a;b]}^{c}=0$
(7.17)

which is known as the second order Bertrand-Darboux equation.

Finally, the system of equations which we have to solve consists of equations
(7.13) - (7.17), where $C_{ab}(q)$ is a KT.

#####  7.2.2 The case of autonomous conservative dynamical systems

We restrict further our considerations to the case of autonomous conservative
dynamical systems so that $V=V(q)$ and $Q^{a}=0$. In this case, the system of
equations (7.13) - (7.17) reduces as follows:

$\displaystyle g_{,t}C_{ab}+fL_{(a;b)}+B_{(a;b)}$ $\displaystyle=$
$\displaystyle 0$ (7.18) $\displaystyle-2gC_{ab}V^{,b}+f_{,t}L_{a}+K_{,a}$
$\displaystyle=$ $\displaystyle 0$ (7.19) $\displaystyle
K_{,t}-fL_{a}V^{,a}-B_{a}V^{,a}$ $\displaystyle=$ $\displaystyle 0$ (7.20)
$\displaystyle
f_{,tt}L_{a}+f(L_{b}V^{,b})_{;a}+(B_{b}V^{,b})_{;a}-2g_{,t}C_{ab}V^{,b}$
$\displaystyle=$ $\displaystyle 0$ (7.21) $\displaystyle
2g\left(C_{[a\left|c\right|}V^{,c}\right)_{;b]}-f_{,t}L_{[a;b]}$
$\displaystyle=$ $\displaystyle 0.$ (7.22)

Obviously, the solution of this system of PDEs is quite involved and requires
the consideration of many cases and subcases. The general solution of the
system is stated in the following theorem (the proof is given in appendix C).

###### Theorem 7.2.1

Assume that the functions $g(t)$ and $f(t)$ are analytic so that they may be
represented by polynomial functions as follows:

$g(t)=\sum^{n}_{k=0}c_{k}t^{k}=c_{0}+c_{1}t+...+c_{n}t^{n}$ (7.23)
$f(t)=\sum^{m}_{k=0}d_{k}t^{k}=d_{0}+d_{1}t+...+d_{m}t^{m}$ (7.24)

where $n,m\in\mathbb{N}$, or may be infinite, and $c_{k},d_{k}\in\mathbb{R}$.
Then, the independent QFIs of an autonomous conservative dynamical system are
the following:

Integral 1.

$I_{1}=-\frac{t^{2}}{2}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+C_{ab}\dot{q}^{a}\dot{q}^{b}+tL_{a}\dot{q}^{a}+\frac{t^{2}}{2}L_{a}V^{,a}+G(q)$

where $C_{ab}$ and $L_{(a;b)}$ are KTs,
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}$ and
$G_{,a}=2C_{ab}V^{,b}-L_{a}$.

Integral 2.

$I_{2}=-\frac{t^{3}}{3}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+t^{2}L_{a}\dot{q}^{a}+\frac{t^{3}}{3}L_{a}V^{,a}-tB_{(a;b)}\dot{q}^{a}\dot{q}^{b}+B_{a}\dot{q}^{a}+tB_{a}V^{,a}$

where $L_{a}$ and $B_{a}$ are such that $L_{(a;b)}$ and $B_{(a;b)}$ are KTs,
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}$ and
$\left(B_{b}V^{,b}\right)_{,a}=-2B_{(a;b)}V^{,b}-2L_{a}$.

Integral 3.

$I_{3}=-e^{\lambda t}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+\lambda e^{\lambda
t}L_{a}\dot{q}^{a}+e^{\lambda t}L_{a}V^{,a}$

where $\lambda\neq 0$, $L_{a}$ is such that $L_{(a;b)}$ is a KT and
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}-\lambda^{2}L_{a}$.

###### Remark 7.2.1

Concerning the Lie bracket between an arbitrary vector field $B^{a}$ and a
gradient vector field $V^{,a}$, we have the following:

$\left[\mathbf{B},\mathbf{\nabla}V\right]^{a}=B^{b}V^{,a}{}_{,b}-V^{,b}B^{a}{}_{,b}=B^{b}V^{,a}{}_{;b}-V^{,b}B^{a}{}_{;b}\implies$
$\left[\mathbf{B},\mathbf{\nabla}V\right]_{a}=B^{b}V_{;ba}-B_{a;b}V^{,b}=\left(B_{b}V^{,b}\right)_{;a}-2B_{(a;b)}V^{,b}.$

Therefore,
$\left[\mathbf{B},\mathbf{\nabla}V\right]_{a}=0\iff\left(B_{b}V^{,b}\right)_{;a}=2B_{(a;b)}V^{,b}$.
If $B^{a}$ is a KV such that $B_{a}V^{,a}=const$, then the vectors $B^{a}$ and
$V^{,a}$ commute (i.e. $\left[\mathbf{B},\mathbf{\nabla}V\right]^{a}=0$).

For easier reference, we collect the QFIs and the LFIs of Theorem 7.2.1 in
Tables LABEL:Table.QFIs.thm1 and LABEL:Table.LFIs.thm1, respectively.

Table 7.1:  The QFIs of Theorem 7.2.1. QFI | Conditions
---|---
$I_{1}=-\frac{t^{2}}{2}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+C_{ab}\dot{q}^{a}\dot{q}^{b}+tL_{a}\dot{q}^{a}+$ $+\frac{t^{2}}{2}L_{a}V^{,a}+G(q)$ |  $C_{ab},L_{(a;b)}$ are KTs, $\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}$, $G_{,a}=2C_{ab}V^{,b}-L_{a}$
$I_{2}=-\frac{t^{3}}{3}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+t^{2}L_{a}\dot{q}^{a}+\frac{t^{3}}{3}L_{a}V^{,a}-$ $-tB_{(a;b)}\dot{q}^{a}\dot{q}^{b}+B_{a}\dot{q}^{a}+tB_{a}V^{,a}$ |  $L_{(a;b)},B_{(a;b)}$ are KTs, $\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}$, $\left(B_{b}V^{,b}\right)_{,a}=-2B_{(a;b)}V^{,b}-2L_{a}$
$I_{3}=e^{\lambda t}\left(-L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+\lambda L_{a}\dot{q}^{a}+L_{a}V^{,a}\right)$ |  $L_{(a;b)}=KT$, $\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}-\lambda^{2}L_{a}$
Table 7.2:  The LFIs of Theorem 7.2.1. LFI | Conditions
---|---
$I_{1}=-tG_{,a}\dot{q}^{a}-\frac{s}{2}t^{2}+G(q)$ |  $G_{,a}=KV$, $G_{,a}V^{,a}=s$
$I_{2}=(t^{2}L_{a}+B_{a})\dot{q}^{a}+\frac{s}{3}t^{3}+tB_{a}V^{,a}$ |  $L_{a},B_{a}$ are KVs, $L_{a}V^{,a}=s$, $\left(B_{b}V^{,b}\right)_{,a}=-2L_{a}$
$I_{3}=e^{\lambda t}\left(\lambda L_{a}\dot{q}^{a}+L_{a}V^{,a}\right)$ |  $L_{a}=KV$, $\left(L_{b}V^{,b}\right)_{,a}=-\lambda^{2}L_{a}$

We note that all the QFIs reduce to LFIs when the KT $K_{ab}$ vanishes.
Moreover, it can be checked that the FIs listed in Theorem 7.2.1 produce all
the potentials, which admit a LFI or a QFI, given in [146] and are due to
point Noether symmetries. Since, as it is shown in Theorem 3.4.1, these FIs
also follow from a gauged velocity-dependent Noether symmetry, we conclude
that _there does not exist a one-to-one correspondence between Noether FIs and
the type of Noether symmetry._ To illustrate this important statement, we give
some examples.

The QFI of the total energy (Hamiltonian)
$E=\frac{1}{2}\gamma_{ab}\dot{q}^{a}\dot{q}^{b}+V(q)$ (case Integral 1 for
$L_{a}=0$ and $C_{ab}=\frac{\gamma_{ab}}{2}$) is generated by the point
Noether symmetry $\Big{(}\xi=1,\eta_{a}=0;f=0\Big{)}$ and, also, by the gauged
generalized Noether symmetry

$\Big{(}\xi=0,\enskip\eta_{a}=-\gamma_{ab}\dot{q}^{b};\enskip
f=-\frac{1}{2}\gamma_{ab}\dot{q}^{a}\dot{q}^{b}+V(q)\Big{)}.$

The QFI $-I_{2}(L_{a}=0)$ for $B_{a}$ be a HV with homothetic factor
$\psi=const$ is generated by the point Noether symmetry $\Big{(}\xi=2\psi
t,\eta_{a}=B_{a};f=ct\Big{)}$ such that $B_{a}V^{,a}+2\psi V+c=0$, where $c$
is an arbitrary constant, and, also, by the gauged generalized Noether
symmetry

$\Big{(}\xi=0,\enskip\eta_{a}=-2t\psi\gamma_{ab}\dot{q}^{b}+B_{a};\enskip
f=-t\psi\gamma_{ab}\dot{q}^{a}\dot{q}^{b}-tB_{a}V^{,a}\Big{)}.$

As a final example, we consider the QFI $-\frac{I_{3}}{\lambda}$ for the
gradient HV $L_{a}=\Phi(q)_{,a}$ where $\Phi_{;ab}=\psi\gamma_{ab}$ with
$\psi=const$. This QFI is generated by the point Noether symmetry

$\Big{(}\xi=\frac{2\psi}{\lambda}e^{\lambda t},\eta_{a}=e^{\lambda
t}\Phi(q)_{,a};f=\lambda e^{\lambda t}\Phi(q)-\frac{c}{\lambda}e^{\lambda
t}\Big{)}$

where $\lambda,c$ are non-zero constants and $\Phi_{,a}V^{,a}=-2\psi
V-\lambda^{2}\Phi+c$, and, also, by the gauged generalized Noether symmetry

$\Big{(}\xi=0,\eta_{a}=-\frac{2\psi}{\lambda}e^{\lambda
t}\gamma_{ab}\dot{q}^{b}+e^{\lambda
t}\Phi_{,a};f=-\frac{\psi}{\lambda}e^{\lambda
t}\gamma_{ab}\dot{q}^{a}\dot{q}^{b}-\frac{e^{\lambda
t}}{\lambda}\Phi_{,a}V^{,a}\Big{)}.$

As a first application of Theorem 7.2.1, we determine in the next section the
QFIs of geodesic equations.

####  7.3 The QFIs of geodesic equations of an $n$-dimensional Riemannian
space

Concerning the FIs of geodesic equations, we have the following well-known
result (see sec. 39 of [147]):

###### Proposition 7.3.1

The geodesic equations in an $n$-dimensional Riemannian manifold $(M,g_{ab})$
admit $m$th-order FIs of the form

$A_{r_{1}...r_{m}}\lambda^{r_{1}}...\lambda^{r_{m}}=const$ (7.25)

where $\lambda^{a}\equiv\dot{q}^{a}$ and $A_{r_{1}...r_{m}}$ is an $m$th-order
KT of the metric $g_{ab}$.

In order to determine the QFIs of the geodesic equations in an $n$-dimensional
Riemannian space with metric $\gamma_{ab}$, we apply Theorem 7.2.1 with $V=0$.
For each case of Theorem 7.2.1, we have the following:

Integral 1. In this case $L_{a}=-G_{,a}$ and the QFI is written

$I_{1}=\frac{t^{2}}{2}G_{;ab}\dot{q}^{a}\dot{q}^{b}+C_{ab}\dot{q}^{a}\dot{q}^{b}-tG_{,a}\dot{q}^{a}+G(q)$

where $C_{ab}$ and $G_{;ab}$ are KTs.

The QFI $I_{1}$ consists of two independent QFIs, which are the following:

$I_{1a}=C_{ab}\dot{q}^{a}\dot{q}^{b}\enskip\text{and}\enskip
I_{1b}=\frac{t^{2}}{2}G_{;ab}\dot{q}^{a}\dot{q}^{b}-tG_{,a}\dot{q}^{a}+G(q).$

Integral 2. Since $V=0$, the condition
$\left(B_{b}V^{,b}\right)_{,a}=-2B_{(a;b)}V^{,b}-2L_{a}$ implies that
$L_{a}=0$. Therefore, the QFI is written

$I_{2}=-tB_{(a;b)}\dot{q}^{a}\dot{q}^{b}+B_{a}\dot{q}^{a}$

where $B_{a}$ is such that $B_{(a;b)}$ is a KT.

Integral 3. Since $V=0$ and $\lambda\neq 0$, the condition
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}-\lambda^{2}L_{a}$ implies
that $L_{a}=0$. Therefore, the QFI $I_{3}=0$.

We collect the above results in Table LABEL:Table.QFIs.geodesics.

Table 7.3:  The QFIs of geodesic equations. QFI | Condition
---|---
$I_{1a}=C_{ab}\dot{q}^{a}\dot{q}^{b}$ |  $C_{ab}=$ KT
$I_{1b}=\frac{t^{2}}{2}G_{;ab}\dot{q}^{a}\dot{q}^{b}-tG_{,a}\dot{q}^{a}+G(q)$ |  $G_{;ab}=$ KT
$I_{2}=-tB_{(a;b)}\dot{q}^{a}\dot{q}^{b}+B_{a}\dot{q}^{a}$ |  $B_{(a;b)}=$ KT

####  7.4 The general Kepler problem $V=-\frac{k}{r^{\ell}}$

The general Kepler problem is a 3d Euclidean dynamical system with kinetic
metric $\delta_{ij}=diag(1,1,1)$ and potential $V=-\frac{k}{r^{\ell}}$, where
$k,\ell$ are non-zero real constants and
$r=(x^{2}+y^{2}+z^{2})^{\frac{1}{2}}$. This dynamical system reduces to the 3d
harmonic oscillator for $k<0$ and $\ell=-2$ (which is the probe dynamical
system for checking the validity of arguments and calculations); and to the
classical Kepler problem considered earlier by Kalotas [143] for $\ell=1$. The
Lagrangian of the system is

$L=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})+\frac{k}{r^{\ell}}$ (7.26)

with equations of motion:

$\ddot{x}=-\frac{\ell k}{r^{\ell+2}}x,\enskip\ddot{y}=-\frac{\ell
k}{r^{\ell+2}}y,\enskip\ddot{z}=-\frac{\ell k}{r^{\ell+2}}z.$ (7.27)

To determine the QFIs of the above dynamical system, we apply Theorem 7.2.1
using the geometric quantities of $E^{3}$ (see sec. 2.9).

Integral 1.

$I_{1}=-\frac{t^{2}}{2}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+C_{ab}\dot{q}^{a}\dot{q}^{b}+tL_{a}\dot{q}^{a}+\frac{t^{2}}{2}L_{a}V^{,a}+G(q)$

where $C_{ab}$, $L_{(a;b)}$ are KTs,
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}$ and
$G_{,a}=2C_{ab}V^{,b}-L_{a}$.

Since $C_{ab}$ and $L_{(a;b)}$ are KTs, the results of sec. 2.9 imply that:

$\displaystyle C_{11}$ $\displaystyle=$
$\displaystyle\frac{a_{6}}{2}y^{2}+\frac{a_{1}}{2}z^{2}+a_{4}yz+a_{5}y+a_{2}z+a_{3}$
$\displaystyle C_{12}$ $\displaystyle=$
$\displaystyle\frac{a_{10}}{2}z^{2}-\frac{a_{6}}{2}xy-\frac{a_{4}}{2}xz-\frac{a_{14}}{2}yz-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z+a_{17}$
$\displaystyle C_{13}$ $\displaystyle=$
$\displaystyle\frac{a_{14}}{2}y^{2}-\frac{a_{4}}{2}xy-\frac{a_{1}}{2}xz-\frac{a_{10}}{2}yz-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z+a_{19}$
$\displaystyle C_{22}$ $\displaystyle=$
$\displaystyle\frac{a_{6}}{2}x^{2}+\frac{a_{7}}{2}z^{2}+a_{14}xz+a_{15}x+a_{12}z+a_{13}$
$\displaystyle C_{23}$ $\displaystyle=$
$\displaystyle\frac{a_{4}}{2}x^{2}-\frac{a_{14}}{2}xy-\frac{a_{10}}{2}xz-\frac{a_{7}}{2}yz-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z+a_{20}$
$\displaystyle C_{33}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{2}x^{2}+\frac{a_{7}}{2}y^{2}+a_{10}xy+a_{11}x+a_{8}y+a_{9}$
$L_{a}=\left(\begin{array}[]{c}-b_{15}y^{2}-b_{11}z^{2}+b_{5}xy+b_{2}xz+2(b_{16}+b_{18})yz+b_{3}x+2b_{4}y+2b_{1}z+b_{6}\\\
-b_{5}x^{2}-b_{8}z^{2}+b_{15}xy-2b_{18}xz+b_{12}yz+2(b_{17}-b_{4})x+b_{13}y+2b_{7}z+b_{14}\\\
-b_{2}x^{2}-b_{12}y^{2}-2b_{16}xy+b_{11}xz+b_{8}yz+2(b_{19}-b_{1})x+2(b_{20}-b_{7})y+b_{9}z+b_{10}\end{array}\right)$

and

$L_{(1;1)}=b_{5}y+b_{2}z+b_{3},\enskip
L_{(1;2)}=-\frac{b_{5}}{2}x-\frac{b_{15}}{2}y+b_{16}z+b_{17},\enskip
L_{(1;3)}=-\frac{b_{2}}{2}x+b_{18}y-\frac{b_{11}}{2}z+b_{19},$
$L_{(2;2)}=b_{15}x+b_{12}z+b_{13},\enskip
L_{(2;3)}=-(b_{16}+b_{18})x-\frac{b_{12}}{2}y-\frac{b_{8}}{2}z+b_{20},\enskip
L_{(3;3)}=b_{11}x+b_{8}y+b_{9}.$

Substituting the above quantities in
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}$ and taking the integrability
conditions $G_{,[ab]}=0$ of the constraint $G_{,a}=2C_{ab}V^{,b}-L_{a}$, we
find:

$a_{16}=a_{18}=0,\enskip(\ell+2)a_{17}=0,\enskip(\ell+2)a_{19}=0,\enskip(\ell+2)a_{20}=0,$
$(\ell-1)a_{2}=0,\enskip(\ell-1)a_{5}=0,\enskip(\ell-1)a_{11}=0,\enskip
a_{2}=a_{12},\enskip a_{5}=a_{8},\enskip a_{11}=a_{15},$
$(\ell+2)(a_{3}-a_{13})=0,\enskip(\ell+2)(a_{3}-a_{9})=0,\enskip
b_{3}=b_{9}=b_{13},\enskip(\ell-2)b_{3}=0$

and
$b_{1}=b_{2}=b_{4}=b_{5}=b_{6}=b_{7}=b_{8}=b_{10}=b_{11}=b_{12}=b_{14}=b_{15}=b_{16}=b_{17}=b_{18}=b_{19}=b_{20}=0$.

The above conditions lead to the following four cases: a) $\ell=-2$ (3d
harmonic oscillator); b) $\ell=1$ (the Kepler problem); c) $\ell=2$ (Newton-
Côtes potential); and d) $\ell\neq-2,1,2$.

a) Case $\ell=-2$.

We have $L_{a}=0$ and
$a_{2}=a_{5}=a_{8}=a_{11}=a_{12}=a_{15}=a_{16}=a_{18}=0$.

Then, the independent components of the KT $C_{ab}$ are

$\displaystyle C_{11}$ $\displaystyle=$
$\displaystyle\frac{a_{6}}{2}y^{2}+\frac{a_{1}}{2}z^{2}+a_{4}yz+a_{3}$
$\displaystyle C_{12}$ $\displaystyle=$
$\displaystyle\frac{a_{10}}{2}z^{2}-\frac{a_{6}}{2}xy-\frac{a_{4}}{2}xz-\frac{a_{14}}{2}yz+a_{17}$
$\displaystyle C_{13}$ $\displaystyle=$
$\displaystyle\frac{a_{14}}{2}y^{2}-\frac{a_{4}}{2}xy-\frac{a_{1}}{2}xz-\frac{a_{10}}{2}yz+a_{19}$
$\displaystyle C_{22}$ $\displaystyle=$
$\displaystyle\frac{a_{6}}{2}x^{2}+\frac{a_{7}}{2}z^{2}+a_{14}xz+a_{13}$
$\displaystyle C_{23}$ $\displaystyle=$
$\displaystyle\frac{a_{4}}{2}x^{2}-\frac{a_{14}}{2}xy-\frac{a_{10}}{2}xz-\frac{a_{7}}{2}yz+a_{20}$
$\displaystyle C_{33}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{2}x^{2}+\frac{a_{7}}{2}y^{2}+a_{10}xy+a_{9}.$

Substituting these components in $G_{,a}=2C_{ab}V^{,b}$ and integrating wrt
each coordinate, we find

$G(x,y,z)=-2k\left(a_{3}x^{2}+a_{13}y^{2}+a_{9}z^{2}+2a_{17}xy+2a_{19}xz+2a_{20}yz\right).$

The QFI is

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle
C_{ab}\dot{q}^{a}\dot{q}^{b}+G(x,y,z)$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{2}(z\dot{x}-x\dot{z})^{2}+\frac{a_{6}}{2}(y\dot{x}-x\dot{y})^{2}+\frac{a_{7}}{2}(z\dot{y}-y\dot{z})^{2}+a_{3}\left(\dot{x}^{2}-2kx^{2}\right)+a_{9}(\dot{z}^{2}-2kz^{2})+$
$\displaystyle+a_{13}(\dot{y}^{2}-2ky^{2})+a_{4}(y\dot{x}-x\dot{y})(z\dot{x}-x\dot{z})+a_{10}(z\dot{x}-x\dot{z})(z\dot{y}-y\dot{z})-$
$\displaystyle-
a_{14}(z\dot{y}-y\dot{z})(y\dot{x}-x\dot{y})+2a_{17}(\dot{x}\dot{y}-2kxy)+2a_{19}(\dot{x}\dot{z}-2kxz)+2a_{20}(\dot{y}\dot{z}-2kyz)$

which consists of the following independent FIs:

$M_{1}=y\dot{z}-z\dot{y},\enskip M_{2}=z\dot{x}-x\dot{z},\enskip
M_{3}=x\dot{y}-y\dot{x},\enskip B_{ij}=\dot{q}_{i}\dot{q}_{j}-2kq_{i}q_{j}$

where $q_{i}=(x,y,z)$, $M_{i}$ are the components of the angular momentum, and
$B_{ij}$ are the components of a symmetric tensor. From these nine autonomous
FIs, the maximum number of functional independent FIs is $2n-1=5$ since $n=3$
is the dimension of the configuration system.

The total energy of the system is written

$H\equiv
E=\frac{1}{2}(B_{11}+B_{22}+B_{33})=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-kr^{2}.$

For $k=-\frac{1}{2}$, the tensor $B_{ij}=\dot{q}_{i}\dot{q}_{j}+q_{i}q_{j}$ is
the Jauch-Hill-Fradkin tensor [148].

The Poisson brackets (PBs) for the components of the angular momentum give the
well-known relation

$\\{M_{a},M_{b}\\}=\varepsilon_{abc}M^{c}$ (7.28)

where $\varepsilon_{abc}$ is the totally-antisymmetric Levi-Civita symbol.
This means that the LFIs $M_{1},M_{2},M_{3}$ are not in involution and, hence,
they cannot be used for Liouville integrability.

However, the 3d harmonic oscillator is integrable because the triplet
$H,M_{a},B_{aa}$ (the index $a$ is fixed) is functionally independent and in
involution, i.e. $\\{H,B_{aa}\\}=$ $\\{H,M_{a}\\}=$ $\\{B_{aa},M_{a}\\}=0$.
The same properties also hold for the triplet $B_{11},B_{22},B_{33}$.

We compute: $\\{M_{1},B_{22}\\}=\\{B_{33},M_{1}\\}=2B_{23}$,
$\\{M_{2},B_{33}\\}=\\{B_{11},M_{3}\\}=2B_{13}$ and
$\\{M_{3},B_{11}\\}=\\{B_{22},M_{3}\\}=2B_{12}$.

Finally, the system of the 3d harmonic oscillator is also superintegrable
because it is integrable and the five FIs $H,L_{1},L_{2},L_{3},B_{aa}$ are
functionally independent.

b) Case $\ell=1$.

We have $L_{a}=0$, $a_{16}=a_{17}=a_{18}=a_{19}=a_{20}=0$, $a_{2}=a_{12}$,
$a_{3}=a_{9}=a_{13}$, $a_{5}=a_{8}$, and $a_{11}=a_{15}$.

Then, the independent components of the KT $C_{ab}$ are

$\displaystyle C_{11}$ $\displaystyle=$
$\displaystyle\frac{a_{6}}{2}y^{2}+\frac{a_{1}}{2}z^{2}+a_{4}yz+a_{5}y+a_{2}z+a_{3}$
$\displaystyle C_{12}$ $\displaystyle=$
$\displaystyle\frac{a_{10}}{2}z^{2}-\frac{a_{6}}{2}xy-\frac{a_{4}}{2}xz-\frac{a_{14}}{2}yz-\frac{a_{5}}{2}x-\frac{a_{11}}{2}y$
$\displaystyle C_{13}$ $\displaystyle=$
$\displaystyle\frac{a_{14}}{2}y^{2}-\frac{a_{4}}{2}xy-\frac{a_{1}}{2}xz-\frac{a_{10}}{2}yz-\frac{a_{2}}{2}x-\frac{a_{11}}{2}z$
$\displaystyle C_{22}$ $\displaystyle=$
$\displaystyle\frac{a_{6}}{2}x^{2}+\frac{a_{7}}{2}z^{2}+a_{14}xz+a_{11}x+a_{2}z+a_{3}$
$\displaystyle C_{23}$ $\displaystyle=$
$\displaystyle\frac{a_{4}}{2}x^{2}-\frac{a_{14}}{2}xy-\frac{a_{10}}{2}xz-\frac{a_{7}}{2}yz-\frac{a_{2}}{2}y-\frac{a_{5}}{2}z$
$\displaystyle C_{33}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{2}x^{2}+\frac{a_{7}}{2}y^{2}+a_{10}xy+a_{11}x+a_{5}y+a_{3}.$

Substituting the above quantities in $G_{,a}=2c_{0}C_{ab}V^{,b}$ and
integrating wrt each coordinate, we find the function
$G(x,y,z)=-\frac{k}{r}(a_{11}x+a_{5}y+a_{2}z+2a_{3})$.

The QFI is

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{2}(z\dot{x}-x\dot{z})^{2}+\frac{a_{6}}{2}(y\dot{x}-x\dot{y})^{2}+\frac{a_{7}}{2}(z\dot{y}-y\dot{z})^{2}+a_{4}(y\dot{x}-x\dot{y})(z\dot{x}-x\dot{z})+$
$\displaystyle+a_{10}(z\dot{x}-x\dot{z})(z\dot{y}-y\dot{z})-a_{14}(z\dot{y}-y\dot{z})(y\dot{x}-x\dot{y})+2a_{3}\left[\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r}\right]+$
$\displaystyle+a_{2}\left[z(\dot{x}^{2}+\dot{y}^{2})-\dot{z}(x\dot{x}+y\dot{y})-\frac{k}{r}z\right]+a_{5}\left[y(\dot{x}^{2}+\dot{z}^{2})-\dot{y}(x\dot{x}+z\dot{z})-\frac{k}{r}y\right]+$
$\displaystyle+a_{11}\left[x(\dot{y}^{2}+\dot{z}^{2})-\dot{x}(y\dot{y}+z\dot{z})-\frac{k}{r}x\right].$

The QFI $I_{1}$ contains the following irreducible FIs:
i. The three components of the angular momentum $M_{1}=y\dot{z}-z\dot{y}$,
$M_{2}=z\dot{x}-x\dot{z}$ and $M_{3}=x\dot{y}-y\dot{x}$. .
ii. The total energy (Hamiltonian) of the system
$E\equiv\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r}$.
iii. The three components of the Runge-Lenz vector

$\displaystyle R_{1}$ $\displaystyle=$ $\displaystyle
x(\dot{y}^{2}+\dot{z}^{2})-\dot{x}(y\dot{y}+z\dot{z})-\frac{k}{r}x$
$\displaystyle R_{2}$ $\displaystyle=$ $\displaystyle
y(\dot{x}^{2}+\dot{z}^{2})-\dot{y}(x\dot{x}+z\dot{z})-\frac{k}{r}y$
$\displaystyle R_{3}$ $\displaystyle=$ $\displaystyle
z(\dot{x}^{2}+\dot{y}^{2})-\dot{z}(x\dot{x}+y\dot{y})-\frac{k}{r}z$

which can be written in the compact form

$R_{i}=(v^{j}v_{j})x_{i}-(x^{j}v_{j})v_{i}-\frac{k}{r}x_{i}$ (7.29)

where $x_{i}=(x,y,z)$ and $v_{i}=\dot{x}_{i}=(\dot{x},\dot{y},\dot{z})$. The
linear combination $\mu^{i}R_{i}$, where $\mu^{i}$ are arbitrary constants, is
the Noether invariant found in [143].

Using the vector identity
$\mathbf{A}\times\left(\mathbf{B}\times\mathbf{C}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C}$,
equation (7.29) takes the well-known vector form

$\mathbf{R}=\mathbf{v}\times(\mathbf{x}\times\mathbf{v})-\frac{k}{r}\mathbf{x}.$
(7.30)

We should point out that the above seven FIs are not all independent because
they are related via the relations $\mathbf{R}\cdot\mathbf{M}=0$ and
$\mathbf{R}^{2}=k^{2}+2E\mathbf{M}^{2}$. From these relations, we deduce that
there exist only five independent FIs: the total energy $E$, the three
components of the angular momentum $\mathbf{M}$, and the direction of the
Runge-Lenz vector $\mathbf{R}$.

The Kepler potential is Liouville integrable because the three FIs
$E,M_{a},R_{a}$ are functionally independent and in involution, i.e.
$\\{M_{a},E\\}=0$, $\\{R_{a},E\\}=0$ and $\\{M_{a},R_{a}\\}=0$. It is also
superintegrable because it has dimension $n=3$ and admits $2n-1=5$ independent
FIs.

For the components of the Runge-Lenz vector, we find that
$\\{R_{a},M_{b}\\}=\varepsilon_{abc}R^{c}$ and $\\{R_{a},R_{b}\\}$
$=-2\varepsilon_{abc}M^{c}E$.

c) Case $\ell=2$.

We have: $L_{a}=b_{3}\left(\begin{array}[]{c}x\\\ y\\\ z\\\
\end{array}\right)$, $L_{(a;b)}=b_{3}\delta_{ab}$ and

$C_{11}=\frac{a_{6}}{2}y^{2}+\frac{a_{1}}{2}z^{2}+a_{4}yz+a_{3},\enskip
C_{12}=\frac{a_{10}}{2}z^{2}-\frac{a_{6}}{2}xy-\frac{a_{4}}{2}xz-\frac{a_{14}}{2}yz,$
$C_{13}=\frac{a_{14}}{2}y^{2}-\frac{a_{4}}{2}xy-\frac{a_{1}}{2}xz-\frac{a_{10}}{2}yz,\enskip
C_{22}=\frac{a_{6}}{2}x^{2}+\frac{a_{7}}{2}z^{2}+a_{14}xz+a_{3}$
$C_{23}=\frac{a_{4}}{2}x^{2}-\frac{a_{14}}{2}xy-\frac{a_{10}}{2}xz-\frac{a_{7}}{2}yz,\enskip
C_{33}=\frac{a_{1}}{2}x^{2}+\frac{a_{7}}{2}y^{2}+a_{10}xy+a_{3}.$

Substituting in $G_{,a}=2c_{0}C_{ab}V^{,b}$ and integrating wrt each
coordinate, we find the function
$G(x,y,z)=-\frac{2ka_{3}}{r^{2}}-\frac{b_{3}}{2}r^{2}$.

The QFI is

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle-
b_{3}t^{2}\left[\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r^{2}}\right]+\frac{a_{1}}{2}(z\dot{x}-x\dot{z})^{2}+\frac{a_{6}}{2}(y\dot{x}-x\dot{y})^{2}+\frac{a_{7}}{2}(z\dot{y}-y\dot{z})^{2}+$
$\displaystyle+2a_{3}\left[\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r^{2}}\right]+a_{4}(y\dot{x}-x\dot{y})(z\dot{x}-x\dot{z})+a_{10}(z\dot{x}-x\dot{z})(z\dot{y}-y\dot{z})-$
$\displaystyle-
a_{14}(z\dot{y}-y\dot{z})(y\dot{x}-x\dot{y})+b_{3}t(x\dot{x}+y\dot{y}+z\dot{z})-\frac{b_{3}}{2}r^{2}.$

This QFI contains the three components of the angular momentum, the total
energy $E=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r^{2}}$ of
the resulting system, and also the time-dependent FI

$I_{1a}(\ell=2)=-Et^{2}+t(x\dot{x}+y\dot{y}+z\dot{z})-\frac{r^{2}}{2}.$

d) Case $\ell\neq-2,1,2$.

In this case, the KT $C_{ab}$ is that of the case $\ell=2$, the vector
$L_{a}=0$, and $G(x,y,z)=-\frac{2ka_{3}}{r^{\ell}}$.

The QFI is

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{2}(z\dot{x}-x\dot{z})^{2}+\frac{a_{6}}{2}(y\dot{x}-x\dot{y})^{2}+\frac{a_{7}}{2}(z\dot{y}-y\dot{z})^{2}+2a_{3}\left[\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r^{\ell}}\right]+$
$\displaystyle+a_{4}(y\dot{x}-x\dot{y})(z\dot{x}-x\dot{z})+a_{10}(z\dot{x}-x\dot{z})(z\dot{y}-y\dot{z})-a_{14}(z\dot{y}-y\dot{z})(y\dot{x}-x\dot{y})$

which consists of the three LFIs of the angular momentum and the total energy
$E=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r^{\ell}}$ of the
system.

Integral 2.

$I_{2}=-\frac{t^{3}}{3}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+t^{2}L_{a}\dot{q}^{a}+\frac{t^{3}}{3}L_{a}V^{,a}-tB_{(a;b)}\dot{q}^{a}\dot{q}^{b}+B_{a}\dot{q}^{a}+tB_{a}V^{,a}$

where $L_{a}$ and $B_{a}$ are such that $L_{(a;b)}$ and $B_{(a;b)}$ are KTs,
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}$, and
$\left(B_{b}V^{,b}\right)_{,a}=-2B_{(a;b)}V^{,b}-2L_{a}$.

Since $L_{(a;b)}$ and $B_{(a;b)}$ are KTs, we have the following (see sec.
2.9):

$L_{a}=\left(\begin{array}[]{c}-a_{15}y^{2}-a_{11}z^{2}+a_{5}xy+a_{2}xz+2(a_{16}+a_{18})yz+a_{3}x+2a_{4}y+2a_{1}z+a_{6}\\\
-a_{5}x^{2}-a_{8}z^{2}+a_{15}xy-2a_{18}xz+a_{12}yz+2(a_{17}-a_{4})x+a_{13}y+2a_{7}z+a_{14}\\\
-a_{2}x^{2}-a_{12}y^{2}-2a_{16}xy+a_{11}xz+a_{8}yz+2(a_{19}-a_{1})x+2(a_{20}-a_{7})y+a_{9}z+a_{10}\end{array}\right)$
$B_{a}=\left(\begin{array}[]{c}-b_{15}y^{2}-b_{11}z^{2}+b_{5}xy+b_{2}xz+2(b_{16}+b_{18})yz+b_{3}x+2b_{4}y+2b_{1}z+b_{6}\\\
-b_{5}x^{2}-b_{8}z^{2}+b_{15}xy-2b_{18}xz+b_{12}yz+2(b_{17}-b_{4})x+b_{13}y+2b_{7}z+b_{14}\\\
-b_{2}x^{2}-b_{12}y^{2}-2b_{16}xy+b_{11}xz+b_{8}yz+2(b_{19}-b_{1})x+2(b_{20}-b_{7})y+b_{9}z+b_{10}\end{array}\right)$
$L_{(1;1)}=a_{5}y+a_{2}z+a_{3},\enskip
L_{(1;2)}=-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z+a_{17},\enskip
L_{(1;3)}=-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z+a_{19},$
$L_{(2;2)}=a_{15}x+a_{12}z+a_{13},\enskip
L_{(2;3)}=-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z+a_{20},\enskip
L_{(3;3)}=a_{11}x+a_{8}y+a_{9}$

and

$B_{(1;1)}=b_{5}y+b_{2}z+b_{3},\enskip
B_{(1;2)}=-\frac{b_{5}}{2}x-\frac{b_{15}}{2}y+b_{16}z+b_{17},\enskip
B_{(1;3)}=-\frac{b_{2}}{2}x+b_{18}y-\frac{b_{11}}{2}z+b_{19},$
$B_{(2;2)}=b_{15}x+b_{12}z+b_{13},\enskip
B_{(2;3)}=-(b_{16}+b_{18})x-\frac{b_{12}}{2}y-\frac{b_{8}}{2}z+b_{20},\enskip
B_{(3;3)}=b_{11}x+b_{8}y+b_{9}.$

From the constraint $\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}$, we
find: $(\ell-2)a_{3}=0$, $L_{a}=$ $a_{3}\left(\begin{array}[]{c}x\\\ y\\\ z\\\
\end{array}\right)$, and $L_{(a;b)}=a_{3}\delta_{ab}$.

Replacing in the remaining constraint
$\left(B_{b}V^{,b}\right)_{,a}=-2B_{(a;b)}V^{,b}-2L_{a}$, we obtain
$a_{3}=0\implies$ $L_{a}=0$, $(\ell-2)b_{3}=0$,
$B_{a}=b_{3}\left(\begin{array}[]{c}x\\\ y\\\ z\\\ \end{array}\right)$, and
$B_{(a;b)}=b_{3}\delta_{ab}$.

For $\ell\neq 2$, we have $B_{a}=0$ and the QFI $I_{2}=0$. On the other hand,
for $\ell=2$, we get the non-trivial time-dependent LFI

$I_{2}(\ell=2)=-Et+\frac{1}{2}(x\dot{x}+y\dot{y}+z\dot{z}).$

Integral 3.

$I_{3}=-e^{\lambda t}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+\lambda e^{\lambda
t}L_{a}\dot{q}^{a}+e^{\lambda t}L_{a}V^{,a}$

where $\lambda\neq 0$, $L_{a}$ is such that $L_{(a;b)}$ is a reducible KT, and
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}-\lambda^{2}L_{a}$.

Since $L_{(a;b)}$ is a KT, we have the following:

$L_{a}=\left(\begin{array}[]{c}-a_{15}y^{2}-a_{11}z^{2}+a_{5}xy+a_{2}xz+2(a_{16}+a_{18})yz+a_{3}x+2a_{4}y+2a_{1}z+a_{6}\\\
-a_{5}x^{2}-a_{8}z^{2}+a_{15}xy-2a_{18}xz+a_{12}yz+2(a_{17}-a_{4})x+a_{13}y+2a_{7}z+a_{14}\\\
-a_{2}x^{2}-a_{12}y^{2}-2a_{16}xy+a_{11}xz+a_{8}yz+2(a_{19}-a_{1})x+2(a_{20}-a_{7})y+a_{9}z+a_{10}\end{array}\right)$

and

$L_{(1;1)}=a_{5}y+a_{2}z+a_{3},\enskip
L_{(1;2)}=-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z+a_{17},\enskip
L_{(1;3)}=-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z+a_{19},$
$L_{(2;2)}=a_{15}x+a_{12}z+a_{13},\enskip
L_{(2;3)}=-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z+a_{20},\enskip
L_{(3;3)}=a_{11}x+a_{8}y+a_{9}.$

Substituting in
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}-\lambda^{2}L_{a}$, we find
that the vector $L_{a}$ survives only for $\ell=-2$. Therefore, in what it
follows, we consider only that case.

We compute

$\left(L_{b}V^{,b}\right)_{,a}=-2k\left(\begin{array}[]{c}2a_{3}x+2a_{17}y+2a_{19}z+a_{6}\\\
2a_{13}y+2a_{17}x+2a_{20}z+a_{14}\\\
2a_{9}z+2a_{19}x+2a_{20}y+a_{10}\end{array}\right)$

and

$L_{(a;b)}V^{,b}=-k\left(\begin{array}[]{c}-a_{15}y^{2}-a_{11}z^{2}+a_{5}xy+a_{2}xz+2(a_{16}+a_{18})yz+2a_{3}x+2a_{17}y+2a_{19}z\\\
-a_{5}x^{2}-a_{8}z^{2}+a_{15}xy-2a_{18}xz+a_{12}yz+2a_{17}x+2a_{13}y+2a_{20}z\\\
-a_{2}x^{2}-a_{12}y^{2}-2a_{16}xy+a_{11}xz+a_{8}yz+2a_{19}x+2a_{20}y+2a_{9}z\end{array}\right).$

Therefore, the condition
$\lambda^{2}L_{a}+\left(L_{b}V^{,b}\right)_{,a}+2L_{(a;b)}V^{,b}=0$ gives the
following set of algebraic equations:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-
a_{15}(\lambda^{2}-2k)y^{2}-a_{11}(\lambda^{2}-2k)z^{2}+a_{5}(\lambda^{2}-2k)xy+a_{2}(\lambda^{2}-2k)xz+$
$\displaystyle+2(a_{16}+a_{18})(\lambda^{2}-2k)yz+a_{3}(\lambda^{2}-8k)x+2(\lambda^{2}a_{4}-4ka_{17})y+$
$\displaystyle+2(\lambda^{2}a_{1}-4ka_{19})z+(\lambda^{2}-2k)a_{6}$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle-
a_{5}(\lambda^{2}-2k)x^{2}-a_{8}(\lambda^{2}-2k)z^{2}+a_{15}(\lambda^{2}-2k)xy-2a_{18}(\lambda^{2}-2k)xz+$
$\displaystyle+2a_{12}(\lambda^{2}-2k)yz+2\left[a_{17}(\lambda^{2}-4k)-\lambda^{2}a_{4}\right]x+a_{13}(\lambda^{2}-8k)y+$
$\displaystyle+2(\lambda^{2}a_{7}-4ka_{20})z+(\lambda^{2}-2k)a_{14}$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle-
a_{2}(\lambda^{2}-2k)x^{2}-a_{12}(\lambda^{2}-2k)y^{2}-2a_{16}(\lambda^{2}-2k)xy+a_{11}(\lambda^{2}-2k)xz+$
$\displaystyle+a_{8}(\lambda^{2}-2k)yz+2\left[a_{19}(\lambda^{2}-4k)-\lambda^{2}a_{1}\right]x+2\left[a_{20}(\lambda^{2}-4k)-\lambda^{2}a_{7}\right]y+$
$\displaystyle+a_{9}(\lambda^{2}-8k)z+(\lambda^{2}-2k)a_{10}.$

We consider the following cases:

a) For $\lambda^{2}=2k$, we have
$a_{1}=a_{3}=a_{4}=a_{7}=a_{9}=a_{13}=a_{17}=a_{19}=a_{20}=0$.

Then,

$L_{a}=\left(\begin{array}[]{c}-a_{15}y^{2}-a_{11}z^{2}+a_{5}xy+a_{2}xz+2(a_{16}+a_{18})yz+a_{6}\\\
-a_{5}x^{2}-a_{8}z^{2}+a_{15}xy-2a_{18}xz+a_{12}yz+a_{14}\\\
-a_{2}x^{2}-a_{12}y^{2}-2a_{16}xy+a_{11}xz+a_{8}yz+a_{10}\end{array}\right)$

and

$L_{(a;b)}=\left(\begin{array}[]{ccc}a_{5}y+a_{2}z&-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z&-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z\\\
-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z&a_{15}x+a_{12}z&-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z\\\
-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z&-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z&a_{11}x+a_{8}y\end{array}\right).$

The QFI is

$\displaystyle I_{3a}(\ell=-2)$ $\displaystyle=$
$\displaystyle\left(a_{6}-\frac{a_{2}}{\lambda}M_{2}+\frac{a_{5}}{\lambda}M_{3}\right)e^{\lambda
t}(\dot{x}-\lambda x)+$
$\displaystyle+\left(a_{14}+\frac{a_{12}}{\lambda}M_{1}-\frac{2a_{16}}{\lambda}M_{2}-\frac{a_{15}}{\lambda}M_{3}\right)e^{\lambda
t}(\dot{y}-\lambda y)+$
$\displaystyle+\left(a_{10}-\frac{a_{8}}{\lambda}M_{1}+\frac{a_{11}}{\lambda}M_{2}+\frac{2a_{18}}{\lambda}M_{3}\right)e^{\lambda
t}(\dot{z}-\lambda z).$

From this expression, we have the following irreducible time-dependent LFIs:

$I_{3a1}=e^{\lambda t}(\dot{x}-\lambda x),\enskip I_{3a2}=e^{\lambda
t}(\dot{y}-\lambda y),\enskip I_{3a3}=e^{\lambda t}(\dot{z}-\lambda z).$

If $k>0$, then $\lambda=\pm\sqrt{2k}$; and if $k<0$, then $\lambda=\pm
i\sqrt{-2k}$. Therefore, for all values of the non-zero parameter $k$ of the
system, there exist two constants $\lambda_{\pm}$ each generating three
independent LFIs of the system. We have555The calculations are the same for
either $k>0$ or $k<0$. We continue for $k<0$, which is the case of the 3d
harmonic oscillator.:

$I_{3a1\pm}=e^{\pm i\sqrt{-2k}t}(\dot{x}\mp i\sqrt{-2k}x),\enskip
I_{3a2\pm}=e^{\pm i\sqrt{-2k}t}(\dot{y}\mp i\sqrt{-2k}y),\enskip
I_{3a3\pm}=e^{\pm i\sqrt{-2k}t}(\dot{z}\mp i\sqrt{-2k}z).$

Using the above six LFIs, we can derive all the FIs found in the case Integral
1 for $\ell=-2$. Indeed, we have

$I_{3a1+}I_{3a1-}=B_{11},\enskip I_{3a2+}I_{3a2-}=B_{22},\enskip
I_{3a3+}I_{3a3-}=B_{33},$ $I_{3a1\pm}I_{3a2\mp}=B_{12}\mp
i\sqrt{-2k}M_{3},\enskip I_{3a1\pm}I_{3a3\mp}=B_{13}\pm
i\sqrt{-2k}M_{2},\enskip I_{3a2\pm}I_{3a3\mp}=B_{23}\mp i\sqrt{-2k}M_{1}.$

Therefore, _all the components of the Jauch-Hill-Fradkin tensor $B_{ij}$ can
be constructed by the three components of the angular momentum and the six
time-dependent LFIs $I_{3a1\pm}$, $I_{3a2\pm}$ and $I_{3a3\pm}$_.

b) For $\lambda^{2}=4k$, we get $L_{a}=0$. Therefore, the QFI vanishes.

c) For $\lambda^{2}=8k$, we have
$a_{2}=a_{5}=a_{6}=a_{8}=a_{10}=a_{11}=a_{12}=a_{14}=a_{15}=a_{16}=a_{18}=0$,
$a_{17}=2a_{4}$, $a_{19}=2a_{1}$ and $a_{20}=2a_{7}$.

Then, $L_{a}=\left(\begin{array}[]{c}a_{3}x+a_{17}y+a_{19}z\\\
a_{17}x+a_{13}y+a_{20}z\\\ a_{19}x+a_{20}y+a_{9}z\end{array}\right)$ and
$L_{(a;b)}=\left(\begin{array}[]{ccc}a_{3}&a_{17}&a_{19}\\\
a_{17}&a_{13}&a_{20}\\\ a_{19}&a_{20}&a_{9}\end{array}\right)$.

The QFI is

$\displaystyle I_{3c}(\ell=-2)$
$\displaystyle=-\frac{a_{3}}{\lambda}e^{\lambda
t}\left(\dot{x}-\frac{\lambda}{2}x\right)^{2}-\frac{a_{9}}{\lambda}e^{\lambda
t}\left(\dot{z}-\frac{\lambda}{2}z\right)^{2}-\frac{a_{13}}{\lambda}e^{\lambda
t}\left(\dot{y}-\frac{\lambda}{2}y\right)^{2}-$
$\displaystyle-\frac{a_{17}}{\lambda}e^{\lambda
t}\left[2\dot{x}\dot{y}+\frac{\lambda^{2}}{2}xy-\lambda(y\dot{x}+x\dot{y})\right]-\frac{a_{19}}{\lambda}e^{\lambda
t}\left[2\dot{x}\dot{z}+\frac{\lambda^{2}}{2}xz-\lambda(z\dot{x}+x\dot{z})\right]-$
$\displaystyle-\frac{a_{20}}{\lambda}e^{\lambda
t}\left[2\dot{y}\dot{z}+\frac{\lambda^{2}}{2}yz-\lambda(y\dot{z}+z\dot{y})\right].$

This expression consists of the time-dependent FIs: $I_{3b1}=e^{\lambda
t}\left(\dot{x}-\frac{\lambda}{2}x\right)^{2}$, $I_{3b2}=e^{\lambda
t}\left(\dot{y}-\frac{\lambda}{2}y\right)^{2}$, $I_{3b3}=e^{\lambda
t}\left(\dot{z}-\frac{\lambda}{2}z\right)^{2}$, $I_{3b4}=e^{\lambda
t}\left[\dot{x}\dot{y}+\frac{\lambda^{2}}{4}xy-\frac{\lambda}{2}(y\dot{x}+x\dot{y})\right],\enskip
I_{3b5}=e^{\lambda
t}\left[\dot{x}\dot{z}+\frac{\lambda^{2}}{4}xz-\frac{\lambda}{2}(z\dot{x}+x\dot{z})\right]$
and
$I_{3b6}=e^{\lambda
t}\left[\dot{y}\dot{z}+\frac{\lambda^{2}}{4}yz-\frac{\lambda}{2}(y\dot{z}+z\dot{y})\right]$.

If $k>0$, $\lambda=\pm 2\sqrt{2k}$; and if $k<0$, $\lambda=\pm 2i\sqrt{-2k}$.
Similarly to the calculations of the case a), we find that (we continue for
$k<0$ and adopt the notation of the case a) for the FIs)

$I_{3b1\pm}=(I_{3a1\pm})^{2},\enskip I_{3b2\pm}=(I_{3a2\pm})^{2},\enskip
I_{3b3\pm}=(I_{3a3\pm})^{2},\enskip I_{3b4\pm}=I_{3a1\pm}I_{3a2\pm},$
$I_{3b5\pm}=I_{3a1\pm}I_{3a3\pm},\enskip I_{3b6\pm}=I_{3a2\pm}I_{3a3\pm}.$

Therefore, this case gives again the six time-dependent LFIs $I_{3a1\pm}$,
$I_{3a2\pm}$ and $I_{3a3\pm}$ of the case a).

We collect the results of this section in Table LABEL:Table.QFIs.Kepler.gen,
where $q^{i}=(x,y,z)$.

Table 7.4:  The LFIs/QFIs of the general Kepler problem. $V=-\frac{k}{r^{\ell}}$ | LFIs and QFIs
---|---
$\forall$ $\ell$ |  $M_{1}=y\dot{z}-z\dot{y}$, $M_{2}=z\dot{x}-x\dot{z}$, $M_{3}=x\dot{y}-y\dot{x}$, $H=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r^{\ell}}$
$\ell=-2$ | $B_{ij}=\dot{q}_{i}\dot{q}_{j}-2kq_{i}q_{j}$
$\ell=-2$, $k>0$ | $I_{3a\pm}=e^{\pm\sqrt{2k}t}(\dot{q}_{a}\mp\sqrt{2k}q_{a})$
$\ell=-2$, $k<0$ | $I_{3a\pm}=e^{\pm i\sqrt{-2k}t}(\dot{q}_{a}\mp i\sqrt{-2k}q_{a})$
$\ell=1$ | $R_{i}=(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{k}{r}q_{i}$
$\ell=2$ |  $I_{1}=-Ht^{2}+t(\dot{q}^{i}q_{i})-\frac{r^{2}}{2}$, $I_{2}=-Ht+\frac{1}{2}(\dot{q}^{i}q_{i})$

####  7.5 The time-dependent FIs

As it has been shown, Theorem 7.2.1 produces all LFIs/QFIs of the autonomous
conservative dynamical equations, i.e. the autonomous and the time-dependent
FIs, the latter being equally important as the former. Furthermore, this is
achieved in a way that is independent of the dimension, the signature, and the
curvature of the kinetic metric defined by the kinetic energy/Lagrangian of
the specific dynamical system. On the contrary, the standard methods determine
mainly the autonomous FIs, usually for low degrees of freedom, and consider
principally the ‘usual’ dynamical systems.

The time-dependent FIs can be used to test the integrability of a dynamical
system and, of course, they can be used to obtain the solution of the
dynamical equations in terms of quadratures. As we have seen in sec. 3.3, the
Liouville integrability (see p. 271, sec. 49 in [33]) requires $n$
functionally independent FIs in involution of the form $I(q,p)$. However, it
has been pointed out666See Theorem 1, p.17, chap. II, para. 2 in [34], and
Theorem 3.4 in [35]. that we can also use time-dependent FIs of the form
$I(q,p,t)$ for the same purpose. It is to be noted that both Theorems in [34]
and [35] refer to non-autonomous Hamiltonians $H(q,p,t)$. Moreover, the
usefulness of the time-dependent FIs can be seen from the examples I and II of
sec. VII in [149].

In order to show the use of the time-dependent FIs in the solution of the
dynamical equations, we consider two cases of the general Kepler equations
(7.27) considered in sec. 7.4.

###### Example 7.5.1

In the case of the potential $V=-kr^{2}$ ($\ell=-2$, $k>0$), we found the six
time-dependent LFIs
$I_{3a\pm}=e^{\pm\sqrt{2k}t}(\dot{q}_{a}\mp\sqrt{2k}q_{a})$. We use these LFIs
to obtain the solution of the corresponding dynamical equations. We have the
following:

$\begin{cases}e^{\sqrt{2k}t}(\dot{x}-\sqrt{2k}x)=A_{+}\\\
e^{-\sqrt{2k}t}(\dot{x}+\sqrt{2k}x)=A_{-}\end{cases}\implies\begin{cases}\dot{x}-\sqrt{2k}x=A_{+}e^{-\sqrt{2k}t}\\\
\dot{x}+\sqrt{2k}x=A_{-}e^{\sqrt{2k}t}\end{cases}\implies$
$\dot{x}=\frac{1}{2}\left(A_{+}e^{-\sqrt{2k}t}+A_{-}e^{\sqrt{2k}t}\right)\implies
x(t)=\frac{1}{2}\left(-\frac{A_{+}}{\sqrt{2k}}e^{-\sqrt{2k}t}+\frac{A_{-}}{\sqrt{2k}}e^{\sqrt{2k}t}\right)$

where $A_{\pm}$ are arbitrary constants. Similarly to the other LFIs, we find:

$y(t)=\frac{1}{2}\left(-\frac{B_{+}}{\sqrt{2k}}e^{-\sqrt{2k}t}+\frac{B_{-}}{\sqrt{2k}}e^{\sqrt{2k}t}\right),\enskip
z(t)=\frac{1}{2}\left(-\frac{C_{+}}{\sqrt{2k}}e^{-\sqrt{2k}t}+\frac{C_{-}}{\sqrt{2k}}e^{\sqrt{2k}t}\right)$

where $B_{\pm}$ and $C_{\pm}$ are arbitrary constants.

###### Example 7.5.2

For the case of the 3d harmonic oscillator (i.e. $\ell=-2$, $k<0$), using the
time-dependent LFIs $I_{3a\pm}=e^{\pm i\sqrt{-2k}t}(\dot{q}_{a}\mp
i\sqrt{-2k}q_{a})$, we find (by working as in the Example 7.5.1):

$x(t)=\frac{1}{2}\left(\frac{iD_{+}}{\sqrt{-2k}}e^{-i\sqrt{-2k}t}-\frac{iD_{-}}{\sqrt{-2k}}e^{i\sqrt{-2k}t}\right),\enskip
y(t)=\frac{1}{2}\left(\frac{iE_{+}}{\sqrt{-2k}}e^{-i\sqrt{-2k}t}-\frac{iE_{-}}{\sqrt{-2k}}e^{i\sqrt{-2k}t}\right),$
$z(t)=\frac{1}{2}\left(\frac{iF_{+}}{\sqrt{-2k}}e^{-i\sqrt{-2k}t}-\frac{iF_{-}}{\sqrt{-2k}}e^{i\sqrt{-2k}t}\right)$

where $D_{\pm}$, $E_{\pm}$ and $F_{\pm}$ are arbitrary constants.

####  7.6 Conclusions

As we have seen in previous sections, the FIs play a crucial role in the
solution of the dynamical equations. Therefore, it is important that one has a
systematic method to compute them for a given dynamical system. In Theorem
7.2.1, we have developed such a method for the case of autonomous conservative
dynamical systems. It has been shown that these FIs are closely related to the
KTs and the symmetries of the kinetic metric, which is defined by the kinetic
energy or the Lagrangian of the particular dynamical system.

Finally, from Theorem 7.2.1, it follows that the determination of a LFI/QFI of
an autonomous conservative dynamical system consists of two parts. One part,
which is entirely characteristic of the kinetic metric and it is common to all
dynamical systems sharing the same metric; and a second part, which consists
of constraints which involve in addition the potential defining the specific
dynamical system.

###  Chapter 8 Second order integrable and superintegrable potentials of 2d
autonomous conservative dynamical systems

In this chapter, we determine the integrable and superintegrable potentials
$V(x,y)$ of Newtonian autonomous conservative dynamical systems in the
Euclidean plane $E^{2}$ by applying Theorem 7.2.1. These potentials are widely
known as second order integrable potentials because only FIs up to second
order (i.e. LFIs and QFIs) are used for establishing their integrability.

####  8.1 Introduction

The determination of integrable and superintegrable systems is a topic that is
in continuous investigation. Obviously, a universal method, which computes the
FIs for all types of dynamical equations independently of their complexity and
degrees of freedom, is not available. For this reason, the existing studies
restrict their considerations to flat spaces or spaces of constant curvature
of low dimension (see e.g. [17, 41, 60, 146, 150, 151, 152, 153] and
references therein). The prevailing cases involve the autonomous conservative
dynamical systems with two degrees of freedom and the classification of the
potential functions in integrable and superintegrable.

The problem of finding integrable and superintegrable potentials in $E^{2}$ is
not new. It was raised for the first time by Darboux [150] and Whittaker (see
ch. XII of [60]) who considered the Newtonian autonomous holonomic systems
with two degrees of freedom and determined most potentials $V(x,y)$ for which
the system has an autonomous QFI other than the Hamiltonian (energy).
Additional potentials found much later by G. Thompson [17, 154]. Furthermore,
a comprehensive review of the known integrable and superintegrable 2d
autonomous potentials is given in [30].

In most of the studies mentioned above, Noether’s theorem and the direct
method (see sec. 3.5) were used. However, other approaches have also appeared.
For example, Koenigs in [155] used coordinate transformations in order to
solve the system of equations resulting from the condition $\\{H,I\\}=0$. The
solution of that system of equations gives the general functional form of the
QFIs and the superintegrable free Hamiltonians, that is, the ones which
possess two more QFIs –in addition to the Hamiltonian– which are functionally
independent. Koenigs’s method has been generalized in several works (see e.g.
[70] and references cited therein) for 2d autonomous conservative systems.

In the following sections, Theorem 7.2.1, which is a ‘product’ of the direct
method, is applied to the case of 2d autonomous conservative dynamical systems
in order to determine the integrable and superintegrable potentials that admit
LFIs/QFIs. It is found that the integrable potentials are classified in Class
I and Class II, and that superintegrable potentials exist in both classes. All
potentials together with their QFIs are listed in tables for easy reference.
Moreover, all the results listed in the review paper of [30] as well as in
more recent works (see e.g. [41, 153]) are recovered, while some new ones are
found which admit time-dependent LFIs/QFIs.

####  8.2 The determination of the QFIs

From theorem 7.2.1, it follows that for the determination of the QFIs the
following problems have to be solved:
a. Determine the KTs of order two of the kinetic metric $\gamma_{ab}$.
b. Determine the special subspace of KTs of order two of the form
$C_{ab}=L_{(a;b)}$ where $L^{a}$ is a vector.
c. Determine the KTs satisfying the constraint $G_{,a}=2C_{ab}V^{,b}$.
d. Find all KVs $L_{a}$ of the kinetic metric which satisfy the constraint
$L_{a}V^{,a}=s$ where $s$ is a constant, possibly zero.

We note that constraints a. and b. depend only on the kinetic metric. Because
the kinetic energy is a positive definite non-singular quadratic 2-form, we
can always choose coordinates in which this form reduces either to
$\delta_{ab}$ or to $A(q)\delta_{ab}.$ Since we know the KTs and all the
collineations of a conformally flat metric (of Euclidean or Lorentzian
character) [23], we already have the results for all autonomous (Newtonian or
special relativistic) conservative dynamical systems.

The involvement of the potential function is only in the constraints c. and d.
which also depend on the geometric characteristics of the kinetic metric.
There are two different ways to proceed.

#####  8.2.1 The potential $V\left(q\right)$ is known

In this case, the following procedure is used:
a) Substitute $V$ in the constraints $L_{a}V^{,a}=s$ and
$G_{,a}=2C_{ab}V^{,b}$ and find conditions for the defining parameters of the
vector $L_{a}$ and the KT $C_{ab}$.
b) From these conditions determine $L_{a}$ and $C_{ab}$.
c) Substitute $C_{ab}$ in the constraint $G_{,a}=2C_{ab}V^{,b}$ and find the
function $G(q)$.
d) Using the above results, write the LFI/QFI $I$ in each case and determine
directly from Theorem 3.4.1 the gauged generalized Noether symmetry.
e) Examine if $I$ can be reduced to simpler independent FIs or if it is new.

#####  8.2.2 The potential $V\left(q\right)$ is unknown

In this case, the following algorithm is used:
a) Compute the KTs and the KVs of the kinetic metric.
b) Solve the PDE $L_{a}V^{,a}=s$ or the111 The integrability conditions for
the scalar $G$ are very general PDEs from which one can find only special
solutions by making additional simplifying assumptions (e.g. symmetries)
involving $L_{a}$, $C_{ab}$ and $V(q)$ itself. Therefore, one does not find
the most general solution. For example, in [156], it is required that the QFI
$I$ is axisymmetric, that is, $\phi^{[1]}I=0$, where
$\phi^{i[1]}=-y\partial_{x}+x\partial_{y}-\dot{y}\partial_{\dot{x}}+\dot{x}\partial_{\dot{y}}$
is the first prolongation of the rotation $\phi^{i}=-y\partial x+x\partial y$.
It is proved easily that in this case we have also the constraints
$L_{\phi}K_{a}=0$ and $L_{\phi}K_{ab}=0$. integrability conditions
$G_{,[ab]}=0$ and find the possible potentials $V(q)$.
c) Substitute the potentials and the KTs found in the constraint
$G_{,a}=2C_{ab}V^{,b}$ and compute (if it exists) the function $G(q)$.
d) Write the LFI/QFI $I$ for each potential and determine the gauged
generalized Noether symmetry.
e) Examine if $I$ can be reduced further to simpler independent FIs or if it
is a new FI.

In the following sections, we assume the potential is not given and apply the
second procedure. For that we need the geometric quantities of the 2d
Euclidean plane $E^{2}$, which have been already determined in sec. 2.8.
According to Theorem 7.2.1, these quantities in $E^{2}$ are common to all 2d
Newtonian systems and what changes in each particular case are the constraints
$G_{,a}=2C_{ab}V^{,b}$ and $L_{a}V^{,a}=s$, which determine the potential
$V(q)$.

####  8.3 Computing the potentials and the FIs

The application of Theorem 7.2.1 in the case of $E^{2}$ indicates that there
are three different ways to find potentials that admit QFIs (other than the
Hamiltonian):

1) The constraint $L_{a}V^{,a}=s$, which leads to the PDE

$(b_{1}+b_{3}y)V_{,x}+(b_{2}-b_{3}x)V_{,y}-s=0.$ (8.1)

2) The constraint $G_{,a}=2C_{ab}V^{,b}$, which leads to the second order
Bertrand-Darboux PDE ($G_{,xy}=G_{,yx}$)

$\displaystyle 0$ $\displaystyle=$ $\displaystyle(\gamma xy+\alpha x+\beta
y-C)(V_{,xx}-V_{,yy})+\left[\gamma(y^{2}-x^{2})-2\beta x+2\alpha
y+A-B\right]V_{,xy}-$ (8.2) $\displaystyle-3(\gamma x+\beta)V_{,y}+3(\gamma
y+\alpha)V_{,x}.$

3) The constraint
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}-\lambda^{2}L_{a}$ with222For
$\lambda=0$ this constraint is a subcase of the constraint
$G_{,a}=2C_{ab}V^{,b}$. Hence, only the case $\lambda\neq 0$ must be
considered. $\lambda\neq 0$ and the integrability condition
$\left(L_{b}V^{,b}\right)_{,xy}=\left(L_{b}V^{,b}\right)_{,yx}$, which lead to
the PDEs:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left[-2\beta y^{2}+2\alpha
xy+Ax+(2C-a_{1})y+a_{2}\right]V_{,xx}+\left(-2\alpha x^{2}+2\beta
xy+a_{1}x+\right.$ (8.3)
$\displaystyle\left.+By+a_{3}\right)V_{,xy}+\left[-6\alpha
x+2a_{1}+(2C-a_{1})\right]V_{,y}+3(2\alpha y+A)V_{,x}+$
$\displaystyle+\lambda^{2}\left[-2\beta y^{2}+2\alpha
xy+Ax+(2C-a_{1})y+a_{2}\right]$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle(-2\alpha x^{2}+2\beta xy+a_{1}x+By+a_{3})V_{,yy}+\left[-2\beta
y^{2}+2\alpha xy+Ax+\right.$ (8.4)
$\displaystyle\left.+(2C-a_{1})y+a_{2}\right]V_{,xy}+3(2\beta
x+B)V_{,y}+\left[-6\beta y+2(2C-a_{1})+a_{1}\right]V_{,x}+$
$\displaystyle+\lambda^{2}(-2\alpha x^{2}+2\beta xy+a_{1}x+By+a_{3})$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle(\alpha x+\beta
y-C)(V_{,xx}-V_{,yy})+\left(-2\beta x+2\alpha y+A-B\right)V_{,xy}-3\beta
V_{,y}+3\alpha V_{,x}+$ (8.5) $\displaystyle+\lambda^{2}(3\alpha x-3\beta
y+C-a_{1}).$

For $\alpha=\beta=0$ and $a_{1}=C$, the PDE (8.5) reduces to PDE (8.2).
Therefore, in order to find new potentials, one of these conditions must be
relaxed. This case of finding potentials is the most difficult because the
problem is overdetermined, i.e. we have a system of three PDEs (8.3) - (8.5)
and only one unknown function, the potential $V(x,y)$.

In the following sections, we solve these constraints and find the admitted
potentials which –as a rule– are integrable. Subsequently, we apply Theorem
7.2.1 to each of these potentials in order to compute the corresponding
LFIs/QFIs. From these results, we determine which of these potentials are
integrable and, especially, superintegrable.

####  8.4 The constraint $L_{a}V^{,a}=s$

The constraint $L_{a}V^{,a}=s$ gives the PDE (8.1) which can be solved using
the method of the characteristic equation.

To cover all possible occurrences, we have to consider the following cases: a)
$b_{3}=0$ and $b_{1}\neq 0$ (KVs $\partial_{x}$ and
$\partial_{x},\partial_{y})$; b) $b_{3}=b_{1}=0$ and $b_{2}\neq 0$ (KV
$\partial_{y})$; and c) $b_{3}\neq 0$ (KVs $y\partial_{x}-x\partial_{y}$;
$\partial_{x},y\partial_{x}-x\partial_{y}$; and
$\partial_{y},y\partial_{x}-x\partial_{y})$. For each case the solution is
shown in Table LABEL:Table.LV.pots.

Table 8.1:  Potentials that satisfy the constraint $L_{a}V^{,a}=s$. Case | KV | $V(x,y)$
---|---|---
a | $b_{3}=0,b_{1}\neq 0$ | $\frac{s}{b_{1}}x+F(b_{1}y-b_{2}x)$
b |  $b_{3}=b_{1}=0$, $b_{2}\neq 0$ | $\frac{s}{b_{2}}y+F(x)$
c | $b_{3}\neq 0$ | $\frac{s}{b_{3}}\tan^{-1}\left(\frac{y+\frac{b_{1}}{b_{3}}}{-x+\frac{b_{2}}{b_{3}}}\right)+F(b_{1}y+\frac{b_{3}}{2}y^{2}-b_{2}x+\frac{b_{3}}{2}x^{2})$

We shall refer to the above solutions of Table LABEL:Table.LV.pots as Class I
potentials. In order to determine if these potentials admit QFIs, we apply
Theorem 7.2.1 to the following types of potentials resulting from Table
LABEL:Table.LV.pots:

$V_{1}=cx+F(y-bx),\enskip V_{2}=cy+F(x),\enskip
V_{3}=c\tan^{-1}\left(\frac{y+b_{1}}{-x+b_{2}}\right)+F\left(\frac{x^{2}+y^{2}}{2}+b_{1}y-b_{2}x\right)$

where $c$ and $b$ are arbitrary constants.

#####  8.4.1 The potential $V_{1}=cx+F(y-bx)$

Case a. $b=0$ and $F=\lambda y$.

The potential reduces to $V_{1a}=cx+\lambda y$.

The irreducible FIs are

$L_{11}=\dot{x}+ct,\enskip L_{12}=\dot{y}+\lambda t,\enskip
Q_{11}=\frac{1}{2}\dot{x}^{2}+cx,\enskip Q_{12}=\frac{1}{2}\dot{y}^{2}+\lambda
y.$

We note that $Q_{11}+Q_{12}=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2})+V=H$ is the
Hamiltonian. We compute $\\{Q_{11},Q_{12}\\}=0$ and $\\{L_{11},Q_{11}\\}=-c$.

The FIs $I_{1}=Q_{11}+Q_{12}$, $I_{2}=\lambda
L_{11}-cL_{12}=\lambda\dot{x}-c\dot{y}$ and $I_{3}=Q_{11}$ are functionally
independent and satisfy the relations $\\{I_{1},I_{2}\\}=\\{I_{1},I_{3}\\}=0$
and $\\{I_{2},I_{3}\\}=-c\lambda$. Therefore, the potential $V_{1a}$ is
superintegrable.

We note that the FIs $I_{2}$ and $Q_{11}$ are, respectively, the FIs (3.1.4)
and (3.2.20) of [30].

Case b. $\frac{d^{2}F}{dw^{2}}\neq 0$ and $w\equiv y-bx$.

The irreducible FIs are the following:

$L_{21}=\dot{x}+b\dot{y}+ct,\enskip
L_{22}=t(\dot{x}+b\dot{y})-(x+by)+\frac{c}{2}t^{2},\enskip
Q_{21}=(\dot{x}+b\dot{y})^{2}+2c(x+by).$

For $F(y-bx)=-\frac{1}{2}\lambda^{2}y^{2}$ and $b=0$, we have the potential
$V_{1b}=cx-\frac{1}{2}\lambda^{2}y^{2}$ where $\lambda\neq 0$, which admits
the additional time-dependent FI $L_{23}=e^{\lambda t}(\dot{y}-\lambda y)$.
Observe that in this case thw QFI $Q_{21}$ reduces to
$Q_{e1}=\frac{1}{2}\dot{x}^{2}+cx$ which using the Hamiltonian generates the
QFI $Q_{e2}\equiv$ $H-Q_{e1}=\frac{1}{2}\dot{y}^{2}-$
$\frac{1}{2}\lambda^{2}y^{2}$.

The LFI $L_{21}(c=0)$ is the (3.1.4) of [30].

We compute $\\{H,L_{21}\\}=\frac{\partial L_{21}}{\partial t}=c$ because
$L_{21}$ is a time-dependent FI.

The potential of the case b is integrable because $\\{H,Q_{21}\\}=0$.

Moreover, $\\{H,L_{22}\\}=L_{21}=\frac{\partial L_{22}}{\partial t}$,
$\\{L_{21},L_{22}\\}=1+b^{2}$,
$\\{Q_{21},L_{21}\\}=2c(1+b^{2})=2c\\{L_{21},L_{22}\\}$ and
$\\{Q_{21},L_{22}\\}=2(1+b^{2})L_{21}=2\\{L_{21},L_{22}\\}L_{21}$.

For the special case $V=cx-\frac{1}{2}\lambda^{2}y^{2}$, we have:
$\\{H,L_{23}\\}=\\{Q_{e2},L_{23}\\}=\lambda L_{23}=\frac{\partial
L_{23}}{\partial t}$ and $\\{Q_{e1},L_{23}\\}=0$. The triplet
$Q_{e1},Q_{e2},L_{23}$ shows that this potential is superintegrable.

We note that in [30] only the Class II potentials (see sec. 8.5) are examined
for superintegrability (see p.108, eqs. (3.2.34) - (3.2.36) in [30]).

For $c\neq 0$, the potential $V_{1}=cx+F(y-bx)$ is not included in [30]
because the author seeks for autonomous LFIs of the form (3.1.1); in that case
$s=0$.

#####  8.4.2 The potential $V_{2}=cy+F(x)$

We consider the case $F^{\prime\prime}=\frac{d^{2}F}{dx^{2}}\neq 0$;
otherwise, we retrieve the potential $V_{1a}$ discussed above.

The irreducible FIs are

$L_{31}=\dot{y}+ct,\enskip Q_{31}=\frac{1}{2}\dot{x}^{2}+F(x),\enskip
Q_{32}=\frac{1}{2}\dot{y}^{2}+cy.$

Therefore, the potential $V_{2}$ is integrable. We note that $V_{2}$ is of the
separable form $V=F_{1}(x)+F_{2}(y)$ (see eq. (3.2.20) of [30]).

For $F(x)=-\frac{1}{2}\lambda^{2}x^{2}$, we obtain the potential
$V_{2a}=cy-\frac{1}{2}\lambda^{2}x^{2}$ with $\lambda\neq 0$, which admits the
additional LFI $L_{32}=e^{\lambda t}(\dot{x}-\lambda x)$. This potential is
superintegrable due to the functionally independent triplet $Q_{31}$, $Q_{32}$
and $L_{32}$.

#####  8.4.3 The potential
$V_{3}=c\tan^{-1}\left(\frac{y+b_{1}}{-x+b_{2}}\right)+F\left(\frac{x^{2}+y^{2}}{2}+b_{1}y-b_{2}x\right)$

We find the time-dependent LFI
$L_{51}=y\dot{x}-x\dot{y}+b_{1}\dot{x}+b_{2}\dot{y}+ct$.

For $c=0$ the considered potential is integrable, while for $c\neq 0$ we do
not know (needs further investigation).

\- For $c=0$ and $F=\lambda\left(\frac{x^{2}+y^{2}}{2}+b_{1}y-b_{2}x\right)$
where $\lambda\neq 0$, the irreducible FIs are:
$L_{41}=y\dot{x}-x\dot{y}+b_{1}\dot{x}+b_{2}\dot{y}$,
$Q_{41}=\frac{1}{2}\dot{x}^{2}+\frac{1}{2}\lambda x^{2}-\lambda b_{2}x$,
$Q_{42}=\frac{1}{2}\dot{y}^{2}+\frac{1}{2}\lambda y^{2}+\lambda b_{1}y$ and
$Q_{43}=\dot{x}\dot{y}+\lambda(xy+b_{1}x-b_{2}y)$.

Observe that $Q_{41}+Q_{42}=H$ is the energy of the system. The LFI $L_{41}$
is the (3.1.6) of [30]. The functionally independent triplet $H$, $L_{41}$,
$Q_{41}$ proves that the considered potential is superintegrable. We compute:
$\\{H,L_{41}\\}=\\{H,Q_{41}\\}=0$ and $\\{L_{41},Q_{41}\\}=-Q_{43}+\lambda
b_{1}b_{2}$.

If $b_{1}=b_{2}=0$ and $\lambda=-k^{2}\neq 0$, we obtain the
superintegrable333 A subcase of the above superintegrable potential is the
potential $V_{3a}=\lambda\left(\frac{x^{2}+y^{2}}{2}+b_{1}y-b_{2}x\right)$.
potential $V_{3b}=-\frac{1}{2}k^{2}(x^{2}+y^{2})$ which admits the additional
time-dependent LFIs

$L_{42\pm}=e^{\pm kt}(\dot{x}\mp kx)\enskip\text{and}\enskip L_{43\pm}=e^{\pm
kt}(\dot{y}\mp ky).$

We also compute: $\\{L_{41},Q_{42}\\}=Q_{43}-\lambda b_{1}b_{2}$,
$\\{L_{41},Q_{43}\\}=2Q_{41}-2Q_{42}+\lambda(b_{2}^{2}-b_{1}^{2})$,
$\\{Q_{41},Q_{42}\\}=0$ and $\\{Q_{41},Q_{43}\\}=\\{Q_{43},Q_{42}\\}=-\lambda
L_{41}$.

In sec. 4 of [157], the author has found the superintegrable Class I
potentials $V_{1a}$ and $V_{3a}$.

We note that in the review [30] the time-dependent LFIs of the potentials
$V_{1a}$ and $V_{2}$ are not discussed. In general, in [30], all the time-
dependent FIs are ignored, although they can be used to decide the
superintegrability of the system.

#####  8.4.4 Summary

We collect the results for the Class I potentials in the Tables
LABEL:Table.Class1.1 and LABEL:Table.Class1.2.

Table 8.2:  Integrable Class I potentials and the special non-integrable potential $V_{3}(c\neq 0)$. Potential | Ref. [30] | LFIs and QFIs
---|---|---
$V_{3}(c\neq 0)=c\tan^{-1}\left(\frac{y+b_{1}}{-x+b_{2}}\right)+F\left(\frac{x^{2}+y^{2}}{2}+b_{1}y-b_{2}x\right)$ | - | $L_{51}=y\dot{x}-x\dot{y}+b_{1}\dot{x}+b_{2}\dot{y}+ct$
Integrable potentials
$V_{1}=cx+F(y-bx)$, $\frac{d^{2}F}{dw^{2}}\neq 0$, $w\equiv y-bx$ | - |  $L_{21}=\dot{x}+b\dot{y}+ct$, $L_{22}=t(\dot{x}+b\dot{y})-(x+by)+\frac{c}{2}t^{2}$, $Q_{21}=(\dot{x}+b\dot{y})^{2}+2c(x+by)$
$V_{2}=cy+F(x)$, $F^{\prime\prime}\neq 0$ | (3.2.20) |  $L_{31}=\dot{y}+ct$, $Q_{31}=\frac{1}{2}\dot{x}^{2}+F(x)$, $Q_{32}=\frac{1}{2}\dot{y}^{2}+cy$
$V_{3}(c=0)$ | (3.1.6) | $L_{51}(c=0)$
Table 8.3:  Superintegrable Class I potentials. Superintegrable potentials
---
Potential | Ref. [30] | LFIs and QFIs
$V_{1a}=cx+\lambda y$ |  (3.1.4), (3.2.20) |  $L_{11}=\dot{x}+ct$, $L_{12}=\dot{y}+\lambda t$, $Q_{11}=\frac{1}{2}\dot{x}^{2}+cx$, $Q_{12}=\frac{1}{2}\dot{y}^{2}+\lambda y$
$V_{1b}=cx-\frac{1}{2}\lambda^{2}y^{2}$, $\lambda\neq 0$ | (3.2.20) |  $L_{11}$, $L_{22}(b=0)=t\dot{x}-x+\frac{c}{2}t^{2}$, $L_{23}=e^{\lambda t}(\dot{y}-\lambda y)$, $Q_{2e1}=\frac{1}{2}\dot{x}^{2}+cx$, $Q_{2e2}=\frac{1}{2}\dot{y}^{2}-\frac{1}{2}\lambda^{2}y^{2}$
$V_{2a}=cy-\frac{1}{2}\lambda^{2}x^{2}$, $\lambda\neq 0$ | (3.2.20) |  $L_{31}$, $Q_{31a}=\frac{1}{2}\dot{x}^{2}-\frac{1}{2}\lambda^{2}x^{2}$, $Q_{32}$, $L_{32}=e^{\lambda t}(\dot{x}-\lambda x)$
$V_{3a}=\lambda\left(\frac{x^{2}+y^{2}}{2}+b_{1}y-b_{2}x\right)$ where $\lambda\neq 0$ | (3.1.6) |  $L_{41}=y\dot{x}-x\dot{y}+b_{1}\dot{x}+b_{2}\dot{y}$, $Q_{41}=\frac{1}{2}\dot{x}^{2}+\frac{1}{2}\lambda x^{2}-\lambda b_{2}x$, $Q_{42}=\frac{1}{2}\dot{y}^{2}+\frac{1}{2}\lambda y^{2}+\lambda b_{1}y$, $Q_{43}=\dot{x}\dot{y}+\lambda(xy+b_{1}x-b_{2}y)$
$V_{3b}=-\frac{1}{2}k^{2}(x^{2}+y^{2})$, $k\neq 0$ | (3.1.5) |  $L_{41b}=y\dot{x}-x\dot{y}$, $Q_{41b}=\frac{1}{2}\dot{x}^{2}-\frac{1}{2}k^{2}x^{2}$, $Q_{42b}=\frac{1}{2}\dot{y}^{2}-\frac{1}{2}k^{2}y^{2}$, $Q_{43b}=\dot{x}\dot{y}-k^{2}xy$, $L_{42\pm}=e^{\pm kt}(\dot{x}\mp kx)$, $L_{43\pm}=e^{\pm kt}(\dot{y}\mp ky)$

We note that the potentials $V_{1}$ and $V_{2}$ in Table LABEL:Table.Class1.1
are also superintegrable due to additional time-dependent LFIs.

####  8.5 The constraint $G_{,a}=2C_{ab}V^{,b}$

In this case, we have the PDE (8.2)

$\displaystyle 0$ $\displaystyle=$ $\displaystyle(\gamma xy+\alpha x+\beta
y-C)(V_{,xx}-V_{,yy})+\left[\gamma(y^{2}-x^{2})-2\beta x+2\alpha
y+A-B\right]V_{,xy}-$ (8.6) $\displaystyle-3(\gamma x+\beta)V_{,y}+3(\gamma
y+\alpha)V_{,x}.$

The potentials which follow from this PDE we call Class II potentials. This
PDE cannot be solved in full generality (see also [30]); therefore, we
consider various cases which produce the known QFIs using Theorem 7.2.1. We
emphasize that the potentials we find in this section are only a subset of the
possible potentials which will follow from the general solution of (8.6).
However, the important point here is that we recover the known results with a
direct and unified approach which can be used in the future by other authors
to discover new integrable and superintegrable potentials in $E^{2}$ and in
other spaces.

1) $\gamma\neq 0$, $A=B$ and $\alpha=\beta=C=0$. Then,
$C_{ab}=\left(\begin{array}[]{cc}\gamma y^{2}+A&-\gamma xy\\\ -\gamma
xy&\gamma x^{2}+A\end{array}\right)$ and equation (8.6) becomes

$xy(V_{,xx}-V_{,yy})+(y^{2}-x^{2})V_{,xy}-3xV_{,y}+3yV_{,x}=0.$ (8.7)

The solution of (8.7) is the potential

$V_{21}=\frac{F_{1}\left(\frac{y}{x}\right)}{d_{1}x^{2}+d_{2}y^{2}}+F_{2}(x^{2}+y^{2})$
(8.8)

where $d_{1}$ and $d_{2}$ are arbitrary constants.

\- For the subcase $d_{1}=d_{2}=1$ with $A=0$, we find the QFI

$I_{11}=(y\dot{x}-x\dot{y})^{2}+2F_{1}\left(\frac{y}{x}\right)=(r^{2}\dot{\theta})^{2}-\Phi(\theta)$
(8.9)

where $r^{2}=x^{2}+y^{2}$ and $\theta=\tan^{-1}\left(\frac{y}{x}\right)$. This
is the well-known Ermakov-Lewis invariant (see eq. (3.2.11) of [30]).

\- For $d_{1}\neq 0$, the potential (8.8) is written equivalently as
$V_{21}=\frac{F_{1}\left(\frac{y}{x}\right)}{x^{2}+cy^{2}}+F_{2}(x^{2}+y^{2})$
where $c$ is an arbitrary constant.

This potential admits QFIs for
$F_{1}=\frac{cky^{2}+kx^{2}}{x^{2}+(2-c)y^{2}}$. Therefore,

$V_{21a}=\frac{k}{x^{2}+(2-c)y^{2}}+F_{2}(x^{2}+y^{2})=\frac{k}{x^{2}+\ell
y^{2}}+F_{2}(x^{2}+y^{2})$

and the associated QFI is

$I_{11a}=(y\dot{x}-x\dot{y})^{2}+\frac{2k(c-1)y^{2}}{x^{2}+(2-c)y^{2}}=(y\dot{x}-x\dot{y})^{2}+\frac{2k(1-\ell)y^{2}}{x^{2}+\ell
y^{2}}$ (8.10)

where $\ell\equiv 2-c$.

\- For $d_{1}=0$ and $d_{2}\neq 0$, the potential (8.8) becomes
$V_{21}=\frac{F_{1}\left(\frac{y}{x}\right)}{y^{2}}+F_{2}(x^{2}+y^{2})$.

This potential admits QFIs for $F_{1}=\frac{ky^{2}}{2x^{2}+y^{2}}$. Then,
$V_{21b}=\frac{k}{2x^{2}+y^{2}}+F(x^{2}+y^{2})$ and the associated QFI is
$I_{11b}=(y\dot{x}-x\dot{y})^{2}+\frac{ky^{2}}{2x^{2}+y^{2}}$. Observe that
$V_{21b}$ is of the form $V_{21a}(c=3/2)$ or $V_{21a}(\ell=1/2)$ with
$\bar{k}\equiv 2k$. Therefore, $V_{21b}$ is included in case $V_{21a}$.

2) $\gamma=1$ and $\alpha=\beta=B=C=0$. Then,
$C_{ab}=\left(\begin{array}[]{cc}y^{2}+A&-xy\\\ -xy&x^{2}\end{array}\right)$
and equation (8.6) becomes

$xy(V_{,xx}-V_{,yy})+(y^{2}-x^{2}+A)V_{,xy}-3xV_{,y}+3yV_{,x}=0.$ (8.11)

\- For $A=0$ equation (8.11) reduces to (8.7).

\- For $A\neq 0$ the PDE (8.11) gives the Darboux solution

$V_{22}=\frac{F_{1}(u)-F_{2}(v)}{u^{2}-v^{2}}$ (8.12)

where $r^{2}=x^{2}+y^{2}$,
$u^{2}=r^{2}+A+\left[(r^{2}+A)^{2}-4Ax^{2}\right]^{1/2}$ and
$v^{2}=r^{2}+A-\left[(r^{2}+A)^{2}-4Ax^{2}\right]^{1/2}$.

We find the QFI (see eq. (3.2.9) of [30]).

$I_{21}=(y\dot{x}-x\dot{y})^{2}+A\dot{x}^{2}+\frac{v^{2}F_{1}(u)-u^{2}F_{2}(v)}{u^{2}-v^{2}}.$
(8.13)

3) $\gamma=1$, $B=-A$, $C=\pm iA\neq 0$ and $\alpha=\beta=0$. Then,

$C_{ab}=\left(\begin{array}[]{cc}y^{2}+A&-xy\pm iA\\\ -xy\pm
iA&x^{2}-A\end{array}\right)$

and equation (8.6) gives again a potential of the form (8.12), but with
$u^{2}=r^{2}+\left[r^{4}-4A(x\pm iy)^{2}\right]^{1/2}$ and
$v^{2}=r^{2}-\left[r^{4}-4A(x\pm iy)^{2}\right]^{1/2}$.

We find the QFI (see eq. (3.2.13) of [30])

$I_{31}=(y\dot{x}-x\dot{y})^{2}+A(\dot{x}\pm
i\dot{y})^{2}+\frac{v^{2}F_{1}(u)-u^{2}F_{2}(v)}{u^{2}-v^{2}}.$ (8.14)

4a) $\alpha=1$ and $\beta=\gamma=A=B=C=0$. Then,

$C_{ab}=\left(\begin{array}[]{cc}2y&-x\\\ -x&0\end{array}\right)$

and equation (8.6) becomes

$x(V_{,xx}-V_{,yy})+2yV_{,xy}+3V_{,x}=0$ (8.15)

which gives the potential

$V_{24}=\frac{F_{1}(r+y)+F_{2}(r-y)}{r}$ (8.16)

where $r^{2}=x^{2}+y^{2}$.

We find the QFI (see eq. (3.2.15) of [30])

$I_{41}=\dot{x}(y\dot{x}-x\dot{y})+\frac{(r+y)F_{2}(r-y)-(r-y)F_{1}(r+y)}{r}.$
(8.17)

4b) $\beta=1$ and $\alpha=\gamma=A=B=C=0$. Then,

$C_{ab}=\left(\begin{array}[]{cc}0&-y\\\ -y&2x\end{array}\right)$

and equation (8.6) becomes

$y(V_{,xx}-V_{,yy})-2xV_{,xy}-3V_{,y}=0$ (8.18)

which gives the potential

$V_{24b}=\frac{F_{1}(r+x)+F_{2}(r-x)}{r}$ (8.19)

where $r^{2}=x^{2}+y^{2}$.

We find the QFI

$I_{41b}=\dot{y}(x\dot{y}-y\dot{x})+\frac{(r+x)F_{2}(r-x)-(r-x)F_{1}(r+x)}{r}.$
(8.20)

###### Remark 8.5.1

Observe that under the rotation $x\leftrightarrow y$ the potential (8.16)
becomes the potential (8.19) and all the results of the case 4b) follow. As a
result, the case 4b) can be overlooked when we assess integrability. This is
not the case for establishing superintegrability where the PDE (8.18) plays a
crucial role (see the superintegrable potential (8.52) in sec. 8.5.2).

5) $\alpha=1$, $\beta=-i$, $A=-B=\frac{i}{4}$, $C=\frac{1}{4}$ and $\gamma=0$.
Then,

$C_{ab}=\left(\begin{array}[]{cc}2y+\frac{i}{4}&-x+iy+\frac{1}{4}\\\
-x+iy+\frac{1}{4}&-2ix-\frac{i}{4}\end{array}\right)$

and equation (8.6) becomes

$(x-iy-\frac{1}{4})\left(V_{,xx}-V_{,yy}\right)+2\left(y+ix+\frac{i}{4}\right)V_{,xy}+3iV_{,y}+3V_{,x}=0.$
(8.21)

This PDE is written equivalently as

$(x-iy)\left(\partial_{x}+i\partial_{y}\right)^{2}V-\frac{1}{4}(\partial_{x}-i\partial_{y})^{2}V+3(\partial_{x}+i\partial_{y})V=0$
(8.22)

which gives the potential

$V_{25}=w^{-1/2}\left[F_{1}(z+\sqrt{w})+F_{2}(z-\sqrt{w})\right]$ (8.23)

where $z=x+iy$ and $w=x-iy$.

We find the QFI (see eq. (3.2.17) of [30])

$I_{51}=(y\dot{x}-x\dot{y})(\dot{x}+i\dot{y})+\frac{i}{8}(\dot{x}-i\dot{y})^{2}+i\left(1-\frac{z}{\sqrt{w}}\right)F_{1}(z+\sqrt{w})+i\left(-1-\frac{z}{\sqrt{w}}\right)F_{2}(z-\sqrt{w}).$
(8.24)

6) $\alpha=1$, $\beta=\mp i$ and $\gamma=A=B=C=0$. Then,

$C_{ab}=\left(\begin{array}[]{cc}2y&-x\pm iy\\\ -x\pm iy&\mp
2ix\end{array}\right)$

and equation (8.6) becomes

$(x\mp iy)\left(V_{,xx}-V_{,yy}\right)+2\left(y\pm ix\right)V_{,xy}\pm
3iV_{,y}+3V_{,x}=0$ (8.25)

from which follows the potential

$V_{26}=\frac{F_{1}(z)}{r}+F_{2}^{\prime}(z)$ (8.26)

where $F_{2}^{\prime}=\frac{dF_{2}}{dz}$ and $z=x\pm iy$.

We find the QFI (see eq. (3.2.18) of [30])

$I_{61}=(y\dot{x}-x\dot{y})(\dot{x}\pm i\dot{y})-izV+iF_{2}(z).$ (8.27)

7) $AB\neq 0$, $A\neq B$ and $\alpha=\beta=\gamma=C=0$. Then,
$C_{ab}=\left(\begin{array}[]{cc}A&0\\\ 0&B\end{array}\right)$ and equation
(8.6) becomes

$(A-B)V_{,xy}=0\implies V_{,xy}=0.$ (8.28)

Solving the PDE (8.28), we find the separable potential

$V_{27}=F_{1}(x)+F_{2}(y)$ (8.29)

where $F_{1}$ and $F_{2}$ are smooth functions of their arguments.

We find the irreducible QFIs (see eq. (3.2.20) of [30]):

$I_{71a}=\dot{x}^{2}+2F_{1}(x)\enskip\text{and}\enskip
I_{71b}=\dot{y}^{2}+2F_{2}(y).$

It can be proved that there are four special potentials of the form (8.29)
which admit additional time-dependent QFIs and, hence, are superintegrable.
These are the following:

7a. The potential
$V_{271}=\frac{k_{1}}{\left(x+c_{1}\right)^{2}}+\frac{k_{2}}{\left(y+c_{2}\right)^{2}}$
which admits the independent QFIs:

$\displaystyle I_{72a}$ $\displaystyle=$
$\displaystyle-\frac{t^{2}}{2}\dot{y}^{2}+t(y+c_{2})\dot{y}-t^{2}\frac{k_{2}}{(y+c_{2})^{2}}-\frac{1}{2}y^{2}-c_{2}y$
$\displaystyle I_{72b}$ $\displaystyle=$
$\displaystyle-\frac{t^{2}}{2}\dot{x}^{2}+t(x+c_{1})\dot{x}-t^{2}\frac{k_{1}}{(x+c_{1})^{2}}-\frac{1}{2}x^{2}-c_{1}x.$

7b. The potential $V_{272}=F_{1}(x)+\frac{k_{2}}{\left(y+c_{2}\right)^{2}}$
which admits the QFI $I_{72a}$.

7c. The potential $V_{273}=F_{2}(y)+\frac{k_{1}}{\left(x+c_{1}\right)^{2}}$
which admits the QFI $I_{72b}$.

7d. The potential (see [61])
$V_{274}=-\frac{\lambda^{2}}{8}(x^{2}+y^{2})-\frac{\lambda^{2}}{4}\left(c_{1}x+c_{2}y\right)-\frac{k_{1}}{(x+c_{1})^{2}}-\frac{k_{2}}{(y+c_{2})^{2}}$
which admits the independent QFIs:

$\displaystyle I_{73a}$ $\displaystyle=$ $\displaystyle e^{\lambda
t}\left[-\dot{x}^{2}+\lambda(x+c_{1})\dot{x}-\frac{\lambda^{2}}{4}(x+c_{1})^{2}+\frac{2k_{1}}{(x+c_{1})^{2}}\right]$
$\displaystyle I_{73b}$ $\displaystyle=$ $\displaystyle e^{\lambda
t}\left[-\dot{y}^{2}+\lambda(y+c_{2})\dot{y}-\frac{\lambda^{2}}{4}(y+c_{2})^{2}+\frac{2k_{2}}{(y+c_{2})^{2}}\right].$

The parameters $\lambda,c_{1},c_{2},k_{1}$, and $k_{2}$ are arbitrary
constants.

8) $C\neq 0$ and $\alpha=\beta=\gamma=0$.

Then, $C_{ab}=\left(\begin{array}[]{cc}A&C\\\ C&B\end{array}\right)$ and
equation (8.6) becomes

$C(V_{,yy}-V_{,xx})+(A-B)V_{,xy}=0.$ (8.30)

Solving equation (8.30), we find the potential

$V_{28}=F_{1}\left(y+b_{0}x+\sqrt{b_{0}^{2}+1}x\right)+F_{2}\left(y+b_{0}x-\sqrt{b_{0}^{2}+1}x\right)$
(8.31)

where $b_{0}\equiv\frac{A-B}{2C}$.

This potential admits the QFI

$I_{81}=A\dot{x}^{2}+B\dot{y}^{2}+2C\dot{x}\dot{y}+(A+B)V+2C\sqrt{b_{0}^{2}+1}(F_{1}-F_{2}).$
(8.32)

We note that $b_{0}(A,B,C)$; therefore, $A,B,C$ are defining parameters of the
potential and, as a result, they do not generate independent QFIs.

For $b_{0}=0$, we have: $A=B$, $V_{,yy}-V_{,xx}=0$ and the potential reduces
to

$V_{28}(b_{0}=0)=F_{1}(y+x)+F_{2}(y-x)$ (8.33)

which is the solution of the 1d wave equation. The QFI (8.32) gives (besides
the Hamiltonian) the independent QFI

$I_{82}=\dot{x}\dot{y}+F_{1}(y+x)-F_{2}(y-x).$

9) $A=2$, $C=\pm i$ and $\alpha=\beta=\gamma=B=0$. Then,

$C_{ab}=\left(\begin{array}[]{cc}2&\pm i\\\ \pm i&0\end{array}\right)$

and equation (8.6) becomes

$\mp i(V_{,xx}-V_{,yy})+2V_{,xy}=0.$ (8.34)

Solving equation (8.34), we find the potential

$V_{29}=r^{2}F_{1}^{\prime\prime}(z)+F_{2}(z)$ (8.35)

where $F_{1}^{\prime\prime}=\frac{d^{2}F_{1}}{dz^{2}}$ and $z=x\pm iy$.

The associated QFI is (see eq. (3.2.21) of [30])

$I_{91}=\dot{x}(\dot{x}\pm i\dot{y})+V_{29}+2zF_{1}^{\prime}(z)-2F_{1}(z).$
(8.36)

10) $A=1$, $B=-1$, $C=i$ and $\alpha=\beta=\gamma=0$.

Then, $C_{ab}=\left(\begin{array}[]{cc}1&i\\\ i&-1\\\ \end{array}\right)$ and
equation (8.6) becomes $i\left(V_{,yy}-V_{,xx}\right)+2V_{,xy}=0$. This PDE
admits the solution

$V_{211}=F_{1}(z)+\bar{z}F_{2}(z)$ (8.37)

where $z=x+iy$ and $F_{1},F_{2}$ are arbitrary complex functions.

The associated QFI is $I_{211}=\dot{z}^{2}+4\int F_{2}(z)dz$.

11) $A=1$, $B=-1$, $C=-i$ and $\alpha=\beta=\gamma=0$.

As in the previous case, we find the potential

$V_{212}=F_{1}(\bar{z})+zF_{2}(\bar{z})$ (8.38)

and the associated QFI $I_{212}=\dot{\bar{z}}^{2}+4\int
F_{2}(\bar{z})d\bar{z}$.

###### Remark 8.5.2

For the trivial KT $C_{ab}=A\delta_{ab}$ in $E^{2}$, where $A$ is an arbitrary
constant, the condition $G_{,a}=2C_{ab}V^{,b}$ implies that $G=2AV$ for all
potentials $V(x,y)$. Therefore, all 2d Newtonian potentials $V(x,y)$ admit the
QFI

$I=A(\dot{x}^{2}+\dot{y}^{2}+2V)=2AH$

where $H$ is the Hamiltonian.

Comparing with previous works, we see that the potentials $V_{21a}$ and
$V_{28}$ are new, while the potential $V_{274}(c_{1}=c_{2}=0)$ is mentioned in
[61].

#####  8.5.1 The potential given in eq. (21) of [64]

The Bertrand-Darboux PDE (8.2) for $\gamma=0$ has been solved in [64] and the
following potential found:

$V(x,y)=\frac{F_{1}\left(\beta x+\alpha
y-\sqrt{D^{2}+E^{2}}\right)+F_{2}\left(\beta x+\alpha
y+\sqrt{D^{2}+E^{2}}\right)}{\sqrt{D^{2}+E^{2}}}$ (8.39)

where $D(x,y)=\alpha x+\beta y-C$, $E(x,y)=-\beta x+\alpha y+\frac{A-B}{2}$
and $D^{2}+E^{2}\neq 0$. We note that for $A=B$, $C=\beta=0$ and $\alpha=1$
the potential (8.39) reduces to the integrable potential (8.16).

The potential (8.39) is integrable if there exists a function $G(x,y)$ such
that $G_{,a}=2C_{ab}V^{,b}$. Then, the associated QFI is

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle(2\alpha
y+A)\dot{x}^{2}+(2\beta x+B)\dot{y}^{2}-2(\alpha x+\beta y-C)\dot{x}\dot{y}+G$
(8.40) $\displaystyle=$ $\displaystyle
2(\beta\dot{y}-\alpha\dot{x})L+A\dot{x}^{2}+2C\dot{x}\dot{y}+B\dot{y}^{2}+G$

where $L\equiv x\dot{y}-y\dot{x}$ is the angular momentum. We note that in eq.
(22) of [64] there is an unnecessary square over the constant $c_{2}$.

#####  8.5.2 The superintegrable potentials

When a potential belongs to two of the above nine Class II cases
simultaneously, it is superintegrable (e.g. the potentials given in eqs.
(3.2.34) - (3.2.36) of [30]) because in that case it admits two additional
autonomous QFIs besides the Hamiltonian. From the results of the section 8.5,
we find the following Class II superintegrable potentials [41, 153]:

S1) The potential (see eq. (3.2.34) of [30], case (b) in [41] and [61])

$V_{s1}=\frac{k}{2}(x^{2}+y^{2})+\frac{b}{x^{2}}+\frac{c}{y^{2}}$ (8.41)

where $k,b,c$ are arbitrary constants.

This potential is of the form (8.8) for $d_{1}=d_{2}=1$,
$F_{1}\left(\frac{y}{x}\right)=b\left(\frac{y}{x}\right)^{2}+c\left(\frac{x}{y}\right)^{2}$
and $F_{2}(x^{2}+y^{2})=\frac{k}{2}(x^{2}+y^{2})+\frac{b+c}{x^{2}+y^{2}}$; and
also of the separable form (8.29). Therefore, $V_{s1}$ admits the additional
QFIs:

$\displaystyle I_{s1a}$ $\displaystyle=$
$\displaystyle(y\dot{x}-x\dot{y})^{2}+2b\frac{y^{2}}{x^{2}}+2c\frac{x^{2}}{y^{2}}$
(8.42) $\displaystyle I_{s1b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{x}^{2}+\frac{k}{2}x^{2}+\frac{b}{x^{2}}$ (8.43)
$\displaystyle I_{s1c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{y}^{2}+\frac{k}{2}y^{2}+\frac{c}{y^{2}}.$ (8.44)

We note that $V_{s1}\left(k=-\frac{\lambda^{2}}{4},b=-k_{1},c=-k_{2}\right)$,
where $\lambda\neq 0$, coincides with the potential $V_{274}$ for
$c_{1}=c_{2}=0$; therefore, it admits also the time-dependent FIs $I_{73a}$
and $I_{73b}$.

S2) Potentials which are both of the form (8.16) and (8.29). Then, we have to
solve the system of the PDEs (8.15) and $V_{,xy}=0$. We find

$V_{s2}=\frac{k_{1}}{2}(x^{2}+4y^{2})+\frac{k_{2}}{x^{2}}+k_{3}y$ (8.45)

and the QFIs:

$\displaystyle I_{s2a}$ $\displaystyle=$
$\displaystyle\dot{x}(y\dot{x}-x\dot{y})-k_{1}yx^{2}+\frac{2k_{2}y}{x^{2}}-\frac{k_{3}}{2}x^{2}$
(8.46) $\displaystyle I_{s2b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{x}^{2}+\frac{k_{1}}{2}x^{2}+\frac{k_{2}}{x^{2}}$
(8.47) $\displaystyle I_{s2c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{y}^{2}+2k_{1}y^{2}+k_{3}y$ (8.48)

where $k_{1},k_{2},k_{3}$ are arbitrary constants.

The potential (8.45) is the superintegrable potential given in the case (a) of
[41]. We note that the QFI $I_{3}^{a}$ in [41] is not correct. The correct is
the QFI $I_{s2a}$ given in equation (8.46).

Moreover, the potential (3.2.35) of [30] is superintegrable only for $b=4a$.
In this case, the resulting potential coincides with $V_{s2}$ for $k_{1}=2a$,
$k_{2}=c$ and $k_{3}=0$.

S3) Potentials which are both of the form (8.8) and (8.16). Then, we have to
solve system of the PDEs (8.7) and (8.15). We find

$V_{s3}=\frac{k_{1}}{x^{2}}+\frac{k_{2}}{r}+\frac{k_{3}y}{rx^{2}}$ (8.49)

and the QFIs:

$\displaystyle I_{s3a}$ $\displaystyle=$
$\displaystyle(y\dot{x}-x\dot{y})^{2}+2k_{1}\frac{y^{2}}{x^{2}}+2k_{3}\frac{ry}{x^{2}}$
(8.50) $\displaystyle I_{s3b}$ $\displaystyle=$
$\displaystyle\dot{x}(y\dot{x}-x\dot{y})+2k_{1}\frac{y}{x^{2}}+k_{2}\frac{y}{r}+k_{3}\frac{x^{2}+2y^{2}}{rx^{2}}$
(8.51)

where $r^{2}=x^{2}+y^{2}$.

The superintegrable potential (8.49) is symmetric ($x\leftrightarrow y$) to
the superintegrable potential of case (c) in [41]. In order to obtain directly
the superintegrable potential of [41], we should consider the form (8.19)
instead of the form (8.16).

We note that if we rename the constants in (8.49) as $k_{1}=b+c$, $k_{2}=a$
and $k_{3}=c-b$, we recover the superintegrable potential (3.2.36) of [30].
Indeed, we have $V_{s3}=\frac{a}{r}+\frac{\frac{b}{r+y}+\frac{c}{r-y}}{r}$.

S4) If we substitute the solution (8.16) of the PDE (8.15) in the PDE (8.18),
we find that both the PDEs (8.15) and (8.18) are satisfied for
$F_{1}(r+y)=k_{1}+k_{2}\sqrt{r+y}$ and $F_{2}(r-y)=k_{3}\sqrt{r-y}$.
Therefore, the potential (see case (d) in [41])

$V_{s4}=\frac{k_{1}}{r}+k_{2}\frac{\sqrt{r+y}}{r}+k_{3}\frac{\sqrt{r-y}}{r}$
(8.52)

is superintegrable with additional QFIs:

$\displaystyle I_{s4a}$ $\displaystyle=$
$\displaystyle\dot{x}(y\dot{x}-x\dot{y})+\frac{k_{1}y}{r}+\frac{k_{3}(r+y)\sqrt{r-y}-k_{2}(r-y)\sqrt{r+y}}{r}$
$\displaystyle I_{s4b}$ $\displaystyle=$
$\displaystyle\dot{y}(x\dot{y}-y\dot{x})+G(x,y)$

provided that there exists a function $G(x,y)$ such that $G_{,x}+yV_{s4,y}=0$
and $G_{,y}+yV_{s4,x}-2xV_{s4,y}=0$.

We note that in the case (d) in [41] the corresponding QFIs $I_{2}^{d}$ and
$I_{3}^{d}$ are not correct, because $\\{H,I_{2}^{d}\\}\neq 0$ and
$\\{H,I_{3}^{d}\\}\neq 0$. Moreover, the superintegrable potential (8.52)
coincides with the potential of the case (E20) in [153]; and it is not
mentioned in the review [30].

S5) If we use the relations $z\bar{z}=r^{2}$ and $z=re^{i\theta}$ where
$\tan\theta=\frac{y}{x}$ and $z=x+iy$, the potentials $V_{21}$ and (8.37) are
equal iff

$V_{21}=V_{211}\implies\frac{A_{1}(\theta)}{r^{2}}+A_{2}(r)=A_{3}(z)+A_{4}(z)r^{2}\implies$
$A_{1}=k_{1}e^{-4i\theta}+k_{2}e^{-2i\theta},\enskip A_{2}=k_{3}r^{2},\enskip
A_{3}=\frac{k_{2}}{z^{2}},\enskip A_{4}=\frac{k_{1}}{z^{4}}+k_{3}$

where $k_{1},k_{2},k_{3}$ are arbitrary constants. Therefore, we find the
superintegrable potential

$V_{s5}=\frac{k_{1}r^{2}}{z^{4}}+\frac{k_{2}}{z^{2}}+k_{3}r^{2}$ (8.53)

with associated QFIs:

$\displaystyle I_{s5a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}M^{2}+k_{1}e^{-4i\theta}+k_{2}e^{-2i\theta}$ (8.54)
$\displaystyle I_{s5b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{z}^{2}-\frac{k_{1}}{z^{2}}+k_{3}z^{2}.$ (8.55)

In Tables LABEL:Table.Class2.1 and LABEL:Table.Class2.2, we collect the
results on Class II potentials together with the corresponding reference to
the review paper [30]. Concerning the notation: $r^{2}=x^{2}+y^{2}$,
$z=x+iy=re^{i\theta}$ and $M_{3}=x\dot{y}-y\dot{x}$ is the angular momentum.

Table 8.4:  Integrable Class II potentials. Integrable potentials
---
Potential | Ref. [30] | LFIs and QFIs
$V_{21}=\frac{F_{1}\left(\frac{y}{x}\right)}{x^{2}+y^{2}}+F_{2}(x^{2}+y^{2})$ | (3.2.10) | $I_{11}=M_{3}^{2}+2F_{1}\left(\frac{y}{x}\right)$
$V_{21a}=\frac{k}{x^{2}+\ell y^{2}}+F_{2}(x^{2}+y^{2})$ | - | $I_{11a}=M_{3}^{2}+\frac{2k(1-\ell)y^{2}}{x^{2}+\ell y^{2}}$
$V_{22}=\frac{F_{1}(u)-F_{2}(v)}{u^{2}-v^{2}}$, $u^{2}=r^{2}+A+\left[(r^{2}+A)^{2}-4Ax^{2}\right]^{1/2}$, $v^{2}=r^{2}+A-\left[(r^{2}+A)^{2}-4Ax^{2}\right]^{1/2}$ | (3.2.7,8) | $I_{21}=M_{3}^{2}+A\dot{x}^{2}+\frac{v^{2}F_{1}(u)-u^{2}F_{2}(v)}{u^{2}-v^{2}}$
$V_{23}=\frac{F_{1}(u)-F_{2}(v)}{u^{2}-v^{2}}$, $u^{2}=r^{2}+\left[r^{4}-4A(x\pm iy)^{2}\right]^{1/2}$, $v^{2}=r^{2}-\left[r^{4}-4A(x\pm iy)^{2}\right]^{1/2}$ | (3.2.7,12) | $I_{31}=M_{3}^{2}+A(\dot{x}\pm i\dot{y})^{2}+\frac{v^{2}F_{1}(u)-u^{2}F_{2}(v)}{u^{2}-v^{2}}$
$V_{24}=\frac{F_{1}(r+y)+F_{2}(r-y)}{r}$ | (3.2.15) | $I_{41}=M_{3}\dot{x}+\frac{(r-y)F_{1}(r+y)-(r+y)F_{2}(r-y)}{r}$
$V_{25}=w^{-1/2}\left[F_{1}(z+\sqrt{w})+F_{2}(z-\sqrt{w})\right]$, $z=x+iy$, $w=x-iy$ | (3.2.17) |  $I_{51}=-M_{3}(\dot{x}+i\dot{y})+\frac{i}{8}(\dot{x}-i\dot{y})^{2}+$ $+i\left(1-\frac{z}{\sqrt{w}}\right)F_{1}(z+\sqrt{w})+$ $+i\left(-1-\frac{z}{\sqrt{w}}\right)F_{2}(z-\sqrt{w})$
$V_{26}=\frac{F_{1}(z)}{r}+F_{2}^{\prime}(z)$, $F_{2}^{\prime}=\frac{dF_{2}}{dz}$, $z=x\pm iy$ | (3.2.18) | $I_{61}=-M_{3}(\dot{x}\pm i\dot{y})-izV+iF_{2}(z)$
$V_{27}=F_{1}(x)+F_{2}(y)$ | (3.2.20) |  $I_{71a}=\frac{1}{2}\dot{x}^{2}+F_{1}(x)$, $I_{71b}=\frac{1}{2}\dot{y}^{2}+F_{2}(y)$
$V_{28}=F_{1}\left(y+b_{0}x+\sqrt{b_{0}^{2}+1}x\right)+$ $+F_{2}\left(y+b_{0}x-\sqrt{b_{0}^{2}+1}x\right)$, $b_{0}\equiv\frac{A-B}{2C}$ | - |  $I_{81}=A\dot{x}^{2}+B\dot{y}^{2}+2C\dot{x}\dot{y}+(A+B)V+$ $+2C\sqrt{b_{0}^{2}+1}(F_{1}-F_{2})$
$V_{28}(b_{0}=0)=F_{1}(y+x)+F_{2}(y-x)$ | - | $I_{82}=\dot{x}\dot{y}+F_{1}(y+x)-F_{2}(y-x)$
$V_{29}=r^{2}F_{1}^{\prime\prime}(z)+F_{2}(z)$, $F_{1}^{\prime\prime}=\frac{d^{2}F_{1}}{dz^{2}}$, $z=x\pm iy$ | (3.2.21) | $I_{91}=\dot{x}(\dot{x}\pm i\dot{y})+V_{29}+2zF_{1}^{\prime}(z)-2F_{1}(z)$
$V_{210}=\frac{F_{1}\left(\beta x+\alpha y-\sqrt{D^{2}+E^{2}}\right)}{\sqrt{D^{2}+E^{2}}}+$ $+\frac{F_{2}\left(\beta x+\alpha y+\sqrt{D^{2}+E^{2}}\right)}{\sqrt{D^{2}+E^{2}}}$ where $D(x,y)=\alpha x+\beta y-C$ and $E(x,y)=-\beta x+\alpha y+\frac{A-B}{2}$ | - |  $I_{210}=2M_{3}(\beta\dot{y}-\alpha\dot{x})+A\dot{x}^{2}+2C\dot{x}\dot{y}+$ $+B\dot{y}^{2}+G(x,y)$ $\frac{G_{,x}}{2}=\left(2\alpha y+A\right)V_{,x}-\left(\alpha x+\beta y-C\right)V_{,y}$ $\frac{G_{,y}}{2}=\left(2\beta x+B\right)V_{,y}-\left(\alpha x+\beta y-C\right)V_{,x}$
$V_{211}=F_{1}(z)+\bar{z}F_{2}(z)$ | - | $I_{211}=\dot{z}^{2}+4\int F_{2}(z)dz$
$V_{212}=F_{1}(\bar{z})+zF_{2}(\bar{z})$ | - | $I_{211}=\dot{\bar{z}}^{2}+4\int F_{2}(\bar{z})d\bar{z}$
Table 8.5:  Superintegrable Class II potentials. Superintegrable potentials
---
Potential | Ref. [30] | LFIs and QFIs
$V_{s1}=\frac{k}{2}(x^{2}+y^{2})+\frac{b}{x^{2}}+\frac{c}{y^{2}}$ | (3.2.34) |  $I_{s1a}=M_{3}^{2}+2b\frac{y^{2}}{x^{2}}+2c\frac{x^{2}}{y^{2}}$, $I_{s1b}=\frac{1}{2}\dot{x}^{2}+\frac{k}{2}x^{2}+\frac{b}{x^{2}}$, $I_{s1c}=\frac{1}{2}\dot{y}^{2}+\frac{k}{2}y^{2}+\frac{c}{y^{2}}$ \- For $k=0$: $I_{72a}$, $I_{72b}$ where $c_{1}=c_{2}=0,k_{1}=b,k_{2}=c$ \- For $k=-\frac{\lambda^{2}}{4}\neq 0$: $I_{73a}$, $I_{73b}$ where $c_{1}=c_{2}=0,k_{1}=-b,k_{2}=-c$
$V_{s2}=\frac{k_{1}}{2}(x^{2}+4y^{2})+\frac{k_{2}}{x^{2}}+k_{3}y$ |  (3.2.35) $k_{3}=0$ |  $I_{s2a}=M_{3}\dot{x}+k_{1}yx^{2}-\frac{2k_{2}y}{x^{2}}+\frac{k_{3}}{2}x^{2}$, $I_{s2b}=\frac{1}{2}\dot{x}^{2}+\frac{k_{1}}{2}x^{2}+\frac{k_{2}}{x^{2}}$, $I_{s2c}=\frac{1}{2}\dot{y}^{2}+2k_{1}y^{2}+k_{3}y$
$V_{s3}=\frac{k_{1}}{x^{2}}+\frac{k_{2}}{r}+\frac{k_{3}y}{rx^{2}}$ | (3.2.36) |  $I_{s3a}=M_{3}^{2}+2k_{1}\frac{y^{2}}{x^{2}}+2k_{3}\frac{ry}{x^{2}}$, $I_{s3b}=-M_{3}\dot{x}+2k_{1}\frac{y}{x^{2}}+k_{2}\frac{y}{r}+k_{3}\frac{x^{2}+2y^{2}}{rx^{2}}$
$V_{s4}=\frac{k_{1}}{r}+k_{2}\frac{\sqrt{r+y}}{r}+k_{3}\frac{\sqrt{r-y}}{r}$ | - |  $I_{s4a}=-M_{3}\dot{x}+\frac{k_{1}y}{r}+$ $+\frac{k_{3}(r+y)\sqrt{r-y}-k_{2}(r-y)\sqrt{r+y}}{r}$, $I_{s4b}=M_{3}\dot{y}+G(x,y)$ where $G_{,x}+yV_{s4,y}=0$ and $G_{,y}+yV_{s4,x}-2xV_{s4,y}=0$
$V_{s5}=\frac{k_{1}r^{2}}{z^{4}}+\frac{k_{2}}{z^{2}}+k_{3}r^{2}$ | - |  $I_{s5a}=\frac{1}{2}M^{2}+k_{1}e^{-4i\theta}+k_{2}e^{-2i\theta}$ $I_{s5b}=\frac{1}{2}\dot{z}^{2}-\frac{k_{1}}{z^{2}}+k_{3}z^{2}$
$V_{271}=\frac{k_{1}}{\left(x+c_{1}\right)^{2}}+\frac{k_{2}}{\left(y+c_{2}\right)^{2}}$ | (3.2.20) |  $I_{71a}$, $I_{71b}$, $I_{72a}=-\frac{t^{2}}{2}\dot{y}^{2}+t(y+c_{2})\dot{y}-t^{2}\frac{k_{2}}{(y+c_{2})^{2}}-$ $-\frac{1}{2}y^{2}-c_{2}y$, $I_{72b}=-\frac{t^{2}}{2}\dot{x}^{2}+t(x+c_{1})\dot{x}-t^{2}\frac{k_{1}}{(x+c_{1})^{2}}-$ $-\frac{1}{2}x^{2}-c_{1}x$
$V_{272}=F_{1}(x)+\frac{k_{2}}{\left(y+c_{2}\right)^{2}}$ | (3.2.20) |  $I_{71a}$, $I_{71b}$, $I_{72a}$
$V_{273}=F_{2}(y)+\frac{k_{1}}{\left(x+c_{1}\right)^{2}}$ | (3.2.20) |  $I_{71a}$, $I_{71b}$, $I_{72b}$
$V_{274}=-\frac{\lambda^{2}}{8}(x^{2}+y^{2})-\frac{\lambda^{2}}{4}\left(c_{1}x+c_{2}y\right)-$ $-\frac{k_{1}}{(x+c_{1})^{2}}-\frac{k_{2}}{(y+c_{2})^{2}}$, $\lambda\neq 0$ | (3.2.20) |  $I_{71a}$, $I_{71b}$, $I_{73a}=e^{\lambda t}\left[-\dot{x}^{2}+\lambda(x+c_{1})\dot{x}-\right.$ $\left.-\frac{\lambda^{2}}{4}(x+c_{1})^{2}+\frac{2k_{1}}{(x+c_{1})^{2}}\right]$, $I_{73b}=e^{\lambda t}\left[-\dot{y}^{2}+\lambda(y+c_{2})\dot{y}-\right.$ $\left.-\frac{\lambda^{2}}{4}(y+c_{2})^{2}+\frac{2k_{2}}{(y+c_{2})^{2}}\right]$

####  8.6 The constraint
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}-\lambda^{2}L_{a}$

The integrability condition of the constraint
$\left(L_{b}V^{,b}\right)_{,a}=-2L_{(a;b)}V^{,b}-\lambda^{2}L_{a}$ gives the
PDE (8.5). As mentioned in section 8.3, in order to find new potentials from
the PDE (8.5) one (or both) of the conditions $\alpha=\beta=0$ and $a_{1}=C$
must be relaxed. However, even if we do find a new potential, this solution
should satisfy also the remaining PDEs (8.3) and (8.4) in order to admit the
time-dependent QFI $I_{3}$ given in case Integral 3 of theorem 7.2.1. New
potentials which admit the QFI $I_{3}$ shall be referred to as Class III
potentials.

We note that the PB $\\{H,I_{3}\\}=\frac{\partial I_{3}}{\partial t}\neq 0$.
Therefore, to find a new integrable potential, we should find a Class III
potential admitting two independent FIs of the form $I_{3}$, say $I_{3a}$ and
$I_{3b}$, such that $\\{I_{3a},I_{3b}\\}=0$.

After relaxing one, or both, of the conditions $\alpha=\beta=0$ and $a_{1}=C$,
we found that the only non-trivial Class III potential is the superintegrable
potential $V_{3b}=-\frac{\lambda^{2}}{2}r^{2}$ (see sec. 8.4.3), which is
derived for $\alpha\neq 0$ or $\beta\neq 0$. Therefore, there are no new Class
III potentials.

####  8.7 Using FIs to find the solution of 2d integrable dynamical systems

In this section, we consider examples which show how one uses the 2d
(super-)integrable potentials to find the solution of the dynamical equations.

1) The superintegrable potential $V_{3b}=-\frac{1}{2}k^{2}(x^{2}+y^{2})$ where
$k\neq 0$.

We find the solution by using the time-dependent LFIs $L_{42\pm}=e^{\pm
kt}(\dot{x}\mp kx)$ and $L_{43\pm}=e^{\pm kt}(\dot{y}\mp ky)$. Specifically,
we have

$\begin{cases}e^{kt}(\dot{x}-kx)=c_{1+}\\\
e^{-kt}(\dot{x}+kx)=c_{1-}\end{cases}\implies\begin{cases}\dot{x}-kx=c_{1+}e^{-kt}\\\
\dot{x}+kx=c_{1-}e^{kt}\end{cases}\implies
x(t)=\frac{c_{1-}}{2k}e^{kt}-\frac{c_{1+}}{2k}e^{-kt}.$

Similarly from the LFIs $L_{43\pm}$, we find

$y(t)=\frac{c_{2-}}{2k}e^{kt}-\frac{c_{2+}}{2k}e^{-kt}$

where $c_{1\pm}$ and $c_{2\pm}$ are arbitrary constants

2) The integrable potential $V_{2}=cy+F(x)$ where $F^{\prime\prime}\neq 0$.

Using the LFI $L_{31}=\dot{y}+ct=c_{1}$, we find directly
$y(t)=-\frac{c}{2}t^{2}+c_{1}t+c_{2}$ where $c,c_{1},c_{2}$ are arbitrary
constants.

Using the QFI $2Q_{31}=\dot{x}^{2}+2F(x)=const=c_{3}$, we get

$\frac{dx}{dt}=\pm\left[-2F(x)+c_{3}\right]^{1/2}\implies
dt=\pm\left[-2F(x)+c_{3}\right]^{-1/2}dx\implies
t=\pm\int\left[-2F(x)+c_{3}\right]^{-1/2}dx+c_{0}$

where $c_{0}$ is an integration constant. The inverse function of $t=t(x)$ is
the solution of the system. If the function $F(x)$ is given, the solution can
be explicitly determined.

3) For the integrable potential $V_{27}=F_{1}(x)+F_{2}(y)$, by using the QFIs

$I_{71a}=\frac{1}{2}\dot{x}^{2}+F_{1}(x)\enskip\text{and}\enskip
I_{71b}=\frac{1}{2}\dot{y}^{2}+F_{2}(y)$

we find:

$t=\int\left[c_{1}-2F_{1}(x)\right]^{-1/2}dx+c_{0},\enskip
t=\int\left[c_{2}-2F_{2}(y)\right]^{-1/2}dy+c_{3}$

where $c_{0},c_{1}=2I_{71a},c_{2}=2I_{71b}$ and $c_{3}$ are arbitrary
constants.

####  8.8 Conclusions

By using Theorem 7.2.1, we have reproduced in a systematic way most integrable
and superintegrable 2d potentials of autonomous conservative Newtonian
dynamical systems. The method used, being covariant, it is directly applicable
to spaces of higher dimensions and to metrics with any signature and
curvature.

We have found two classes of potentials and, in each class, we have determined
the integrable and the superintegrable potentials together with their QFIs. As
the general solution of the PDE (8.2) is not possible, we have found the
potentials due to certain solutions only. New solutions of this PDE will lead
to new integrable and, possibly, superintegrable 2d potentials.

It appears that the most difficult part in the application of Theorem 7.2.1 to
higher dimensions and curved configuration spaces is the determination of the
KTs. The use of algebraic computing is limited once one considers higher
dimensions, since then the number of the components of the KT increases
dramatically. Fortunately, today new techniques in differential geometry have
been developed (see e.g. [16, 23, 24, 158, 159]), especially in the case of
spaces of constant curvature and decomposable spaces, which can help to deal
with this problem.