# Higher order first integrals of autonomous non-Riemannian dynamical systems

Antonios Mitsopoulos1,a), Michael Tsamparlis2,3,b) and Aniekan Ukpong2,3,c)

1Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics,
University of Athens, Panepistemiopolis, Athens 157 83, Greece
2NITheCS, National Institute for Theoretical and Computational Sciences,
Pietermaritzburg 3201, KwaZulu-Natal, South Africa
3TCCMMP, Theoretical and Computational Condensed Matter and Materials Physics
Group,
School of Chemistry and Physics, University of KwaZulu-Natal,
Pietermaritzburg 3201, KwaZulu-Natal, South Africa

a)Author to whom correspondence should be addressed<EMAIL_ADDRESS>
b)Email<EMAIL_ADDRESS>
c)Email<EMAIL_ADDRESS>

###### Abstract

We consider autonomous holonomic dynamical systems defined by equations of the
form $\ddot{q}^{a}=-\Gamma_{bc}^{a}(q)\dot{q}^{b}\dot{q}^{c}$ $-Q^{a}(q)$,
where $\Gamma^{a}_{bc}(q)$ are the coefficients of a symmetric (possibly non-
metrical) connection and $-Q^{a}(q)$ are the generalized forces. We prove a
theorem which for these systems determines autonomous and time-dependent first
integrals (FIs) of any order in a systematic way, using the ‘symmetries’ of
the geometry defined by the dynamical equations. We demonstrate the
application of the theorem to compute linear, quadratic, and cubic FIs of
various Riemannian and non-Riemannian dynamical systems.

Keywords: integrability; non-Riemannian autonomous system; higher order first
integral; symmetric connection; generalized Killing tensor.

##  1 Introduction

A first integral (FI) of a second order set of dynamical equations with
generalized coordinates $q^{a}$ and generalized velocities
$\dot{q}^{a}\equiv\frac{dq^{a}}{dt}$ is a function $I(t,q^{a},\dot{q}^{a})$
satisfying the condition $\frac{dI}{dt}=0$ along the dynamical equations. FIs
are important because they can be used in order to reduce the order of the
dynamical equations and, if they are ‘enough’ in number [1], to find the
solution of the system by quadrature (Liouville integrability).

The standard method to compute the FIs of Lagrangian systems is Noether’s
theorem. A different method is the direct method which requires only the
dynamical equations and was originally introduced by Whittaker [2, 3, 4, 5, 6,
7]. In the latter method, one assumes a functional form for the FI $I$ (e.g. a
polynomial form in $\dot{q}^{a}$) and demands the condition $\frac{dI}{dt}=0$.
Using the dynamical equations to remove the terms $\ddot{q}^{a}$ whenever they
appear, the FI condition leads to a system of partial differential equations
(PDEs) whose solution provides the FIs.

In this work, we apply the direct method to autonomous holonomic dynamical
systems in a space with a symmetric connection $\Gamma_{bc}^{a}(q)$ (not
necessarily Riemannian) which is read from the dynamical equations. We compute
the resulting system of PDEs and solve it in terms of the ‘symmetries’ of
$\Gamma_{bc}^{a}(q)$. The result is stated as Theorem 1 and provides a
systematic method to determine polynomial FIs in velocities of any order,
time-dependent and autonomous, for this type of dynamical systems. In the
special case that the symmetric connection $\Gamma_{bc}^{a}(q)$ is the
Riemannian connection defined by the kinetic metric (kinetic energy)
$\gamma_{ab}(q)$ of the system, the computed FIs are directly related by means
of the Inverse Noether Theorem [8, 9] to gauged generalized (i.e. velocity-
dependent) weak Noether symmetries. Finally, we apply Theorem 1 in order to
find new integrable and superintegrable systems which admit linear (LFIs),
quadratic (QFIs), and cubic FIs (CFIs).

The structure of the paper has as follows. In section 2, using the direct
method, we derive the system of PDEs that must be satisfied by the
coefficients of an $m$th-order FI of an autonomous (in general non-Riemannian)
dynamical system. In section 3, the ‘solution’ of the system of PDEs is stated
as Theorem 1 (the proof of Theorem 1 is given in the appendix). In section 4,
we apply Theorem 1 for $m=1$ in order to consider the LFIs. It is shown that
there are three independent types of LFIs and we determine their explicit
formulae. In section 5, we apply Theorem 1 for $m>1$ and we find that there
are three independent types of higher order FIs. In section 6, we discuss
these independent FIs and provide a procedure by which symmetries of order
smaller than the order of the FI can be removed. Using this procedure, we
remove these lesser symmetries and we find the complete forms of the FIs for
an even order $m=2\nu$ and an odd order $m=2\nu+1$. The results are collected
in Propositions 6, 7, and 8. In section 7, we find a family of two-dimensional
(2d) non-Riemannian autonomous dynamical systems and, by applying Theorem 1,
we determine the LFIs. In sections 8, 9, and 10, we provide further
applications of Theorem 1 for QFIs and CFIs, by extending existing results in
the literature. Finally, in section 11, we draw our conclusions.

##  2 The conditions for $m$th-order FIs

We consider _general_ (i.e. Riemannian and non-Riemannian) autonomous
dynamical systems of the form

$\ddot{q}^{a}=-\Gamma^{a}_{bc}(q)\dot{q}^{b}\dot{q}^{c}-Q^{a}(q)$ (1)

where $q^{a}$ with $a=1,2,...,D$ are the generalized coordinates of the
configuration space of the system, $D$ is the dimension of the configuration
space, a dot over a letter indicates derivation with respect to (wrt) the
parameter $t$ (time) along the trajectory $q^{a}(t)$, Einstein’s summation
convention is applied, $\Gamma^{a}_{bc}(q)$ are the coefficients of a general
connection and $-Q^{a}(q)$ are the generalized forces. Since only the
symmetric part $\Gamma^{a}_{(bc)}$ contributes to the dynamical equations
–without loss of generality– the quantities $\Gamma^{a}_{bc}(q)$ are assumed
to be symmetric.

We look for $m$th-order FIs of the general form

$I^{(m)}=\sum_{r=0}^{m}M_{i_{1}i_{2}...i_{r}}(t,q)\dot{q}^{i_{1}}\dot{q}^{i_{2}}...\dot{q}^{i_{r}}=M+M_{i_{1}}\dot{q}^{i_{1}}+M_{i_{1}i_{2}}\dot{q}^{i_{1}}\dot{q}^{i_{2}}+...+M_{i_{1}i_{2}...i_{m}}\dot{q}^{i_{1}}\dot{q}^{i_{2}}...\dot{q}^{i_{m}}$
(2)

where $M_{i_{1}...i_{r}}(t,q)$ with $r=0,1,...,m$ are totally symmetric
$r$-rank tensors and the index $m\geq 1$ denotes the order of the FI. _We note
that when $r=0$, the quantities $M_{i_{1}...i_{r}}(t,q)$ reduce to the scalar
$M(t,q)$._ For $m=1$, we have the LFIs; for $m=2$, the QFIs; and for $m=3$,
the CFIs.

The FI condition

$\frac{dI^{(m)}}{dt}=0$ (3)

along the dynamical equations (1) results in the following system of PDEs

$M_{i_{1}i_{2}...i_{r},t}+M_{(i_{1}i_{2}...i_{r-1}|i_{r})}-(r+1)M_{i_{1}i_{2}...i_{r}i_{r+1}}Q^{i_{r+1}}=0,\enskip
r=0,1,2,...,m,m+1\\\ $ (4)

which is expanded as follows:

$\displaystyle M_{(i_{1}i_{2}...i_{m}|i_{m+1})}$ $\displaystyle=$
$\displaystyle 0$ (5) $\displaystyle
M_{i_{1}i_{2}...i_{m},t}+M_{(i_{1}i_{2}...i_{m-1}|i_{m})}$ $\displaystyle=$
$\displaystyle 0$ (6) $\displaystyle
M_{i_{1}i_{2}...i_{r},t}+M_{(i_{1}i_{2}...i_{r-1}|i_{r})}-(r+1)M_{i_{1}i_{2}...i_{r}i_{r+1}}Q^{i_{r+1}}$
$\displaystyle=$ $\displaystyle 0,\enskip r=1,2,...,m-1,\enskip m>1$ (7)
$\displaystyle M_{,t}-M_{i_{1}}Q^{i_{1}}$ $\displaystyle=$ $\displaystyle 0.$
(8)

The symbol $|$ denotes the covariant derivative wrt the symmetric connection
$\Gamma^{a}_{bc}$, a comma indicates partial derivative wrt $q^{a}$ or $t$,
round/square brackets indicate symmetrization/antisymmetrization of the
enclosed indices, and indices enclosed between wavy lines are overlooked by
symmetrization or antisymmetrization symbols.

We note that equation (5) is derived from (4) for $r=m+1$; equation (6) from
(4) for $r=m$; and equation (8) from (4) for $r=0$.

Concerning the notation, we remark that:

$M_{i_{1}...i_{r-k}}(r=0)=\begin{cases}M,\enskip k=0\\\ 0,\enskip k\geq
1\end{cases},\enskip M_{i_{1}...i_{r}}(r>m)=0.$

Equations (5) and (6) are purely geometric equations, which are common to all
systems of the form (1) that share the same symmetric connection. In
particular, equation (5) generalizes the concept of Killing tensors (KTs) to a
non-metrical geometry with a symmetric connection $\Gamma^{a}_{bc}$. In this
context, $M_{i_{1}i_{2}...i_{m}}$ is a generalized $\mathbf{m}$th-order KT for
$\Gamma^{a}_{bc}$.

On the other hand, equations (7) and (8) are of a dynamical character, because
they relate the geometric elements with the generalized forces $Q^{a}$ of the
specific dynamical system.

Since the dynamical system (1) is autonomous, we should use the polynomial
method described in [10] in order to solve the system of PDEs (5) - (8).
According to this method, one assumes general polynomial expressions in the
variable $t$ for the tensor quantities $M_{i_{1}...i_{r}}(t,q)$ with
$r=1,2,...,m$ (see eq. (132) in the appendix), and replaces these expressions
in the system of PDEs (5) - (8). Then, the scalar $M(t,q)$ is determined
exactly, and the remaining PDEs reduce to polynomial equations in $t$ whose
coefficients are functions of $q^{a}$. The ‘solution’ of the latter system of
PDEs is stated below as Theorem 1. A detailed proof is given in the appendix.

##  3 Theorem for $m$th-order FIs of a general autonomous dynamical system

###### Theorem 1

There are two types of $m$th-order FIs for the autonomous (in general non-
Riemannian) dynamical system (1). These are the following:

Integral 1.

$I^{(m)}_{n}=\sum_{r=1}^{m}\left(\sum^{n}_{N=0}L_{(N)i_{1}...i_{r}}t^{N}\right)\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+s_{0}\frac{t^{n+1}}{n+1}+\sum^{n>0}_{N=1}L_{(N-1)c}Q^{c}\frac{t^{N}}{N}+G(q)$
(9)

where $m\geq 1$, $n\geq 0$, $L_{(N)i_{1}...i_{m}}(q)$ with $N=0,1,...,n$ are
$\mathbf{m}$th-order generalized KTs satisfying the condition

$L_{(k)i_{1}...i_{m}}=-\frac{1}{k}L_{(k-1)(i_{1}...i_{m-1}|i_{m})},\enskip
k=1,2,...,n,\enskip n>0,\enskip m>1,$ (10)

the totally symmetric tensor $L_{(n)i_{1}...i_{m-1}}(q)$ with $m>1$ is an
$\mathbf{(m-1)}$th-order generalized KT, the constant $s_{0}$ is defined by
the condition

$L_{(n)a}Q^{a}=s_{0}$ (11)

while the function $G(q)$ and the totally symmetric tensors
$L_{(N)i_{1}...i_{r}}(q)$ satisfy the conditions:

$\displaystyle G_{,i_{1}}$ $\displaystyle=$ $\displaystyle
2L_{(0)i_{1}i_{2}}(m>1)Q^{i_{2}}-L_{(1)i_{1}}(n>0)$ (12)
$\displaystyle\left(L_{(k-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle
2kL_{(k)i_{1}i_{2}}(m>1)Q^{i_{2}}-k(k+1)L_{(k+1)i_{1}}(k<n),\enskip
k=1,2,...,n,\enskip n>0$ (13) $\displaystyle L_{(k)(i_{1}...i_{r-1}|i_{r})}$
$\displaystyle=$
$\displaystyle(r+1)L_{(k)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(k+1)L_{(k+1)i_{1}...i_{r}}(k<n),$
(14) $\displaystyle k=0,1,...,n,\enskip r=2,3,...,m-1,\enskip m>2.$

Integral 2.

$I^{(m)}_{e}=\frac{e^{\lambda
t}}{\lambda}\left(\lambda\sum^{m}_{r=1}L_{i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+L_{c}Q^{c}\right)$
(15)

where $\lambda\neq 0$, $L_{i_{1}...i_{m}}(q)$ is an $\mathbf{m}$th-order
generalized KT satisfying the condition

$L_{i_{1}...i_{m}}=-\frac{1}{\lambda}L_{(i_{1}...i_{m-1}|i_{m})},\enskip m>1$
(16)

and the totally symmetric tensors $L_{i_{1}...i_{r}}(q)$ satisfy the
conditions:

$\displaystyle\left(L_{c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle 2\lambda L_{i_{1}i_{2}}(m>1)Q^{i_{2}}-\lambda^{2}L_{i_{1}}$
(17) $\displaystyle L_{(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-\lambda
L_{i_{1}...i_{r}},\enskip r=2,3,...,m-1,\enskip m>2.$ (18)

Concerning the notation, the symbol $I^{(m)}_{n}$ means the $m$th-order FI
(upper index) with degree of time-dependence $n$ (lower index), while
$I^{(m)}_{e}$ indicates the $m$th-order FI with exponential time-dependence
(lower index).

Using mathematical induction, one also proves the following recursion
formulae.

###### Proposition 2

The independent $m$th-order FIs $I_{n}^{(m)}$ and $I^{(m)}_{e}$ satisfy the
following recursion formulae:
a. $I_{n}^{(k)}<I_{n}^{(k+1)}$, that is, each $k$th-order FI $I_{n}^{(k)}$ is
a subcase of the next $(k+1)$th-order FI $I_{n}^{(k+1)}$ with the same degree
$n$ of time-dependence for all integers $k\geq 1$.
b. $I_{\ell}^{(m)}<I_{\ell+1}^{(m)}$, that is, the $m$th-order FI
$I_{\ell}^{(m)}$ with time-dependence fixed by $\ell$ is a subcase of the
$m$th-order FI $I_{\ell+1}^{(m)}$ with time-dependence $\ell+1$ for all
integers $\ell\geq 0$.
c. $I_{e}^{(k)}<I_{e}^{(k+1)}$, that is, each $k$th-order FI $I_{e}^{(k)}$ is
a subcase of the next $(k+1)$th-order FI $I_{e}^{(k+1)}$ for all integers
$k\geq 1$.

We note that Theorem 1 for $m=2$ (QFIs) and a Riemannian connection reduces to
Theorem 3 of [9].

In the case of a Riemannian connection, by means of the Inverse Noether
Theorem [8, 9], the general $m$th-order FIs (2) are related to the generalized
gauged weak Noether symmetry

$\left(\xi=0,\enskip\eta_{i_{1}}=-\frac{\partial
I^{(m)}}{\partial\dot{q}^{i_{1}}},\enskip\phi_{a},\enskip
f=I^{(m)}-\frac{\partial
I^{(m)}}{\partial\dot{q}^{i_{1}}}\dot{q}^{i_{1}}\right)\quad\text{such that
$\phi_{a}\dot{q}^{a}+F^{a}\frac{\partial I^{(m)}}{\partial\dot{q}^{a}}=0$}$
(19)

where $F^{a}(t,q,\dot{q})$ are the non-conservative generalized forces,
$\phi^{a}(t,q,\dot{q})$ is an additional vector generator, $f(t,q,\dot{q})$ is
the Noether function,
$\mathbf{X}=\xi(t,q,\dot{q})\partial_{t}+\eta^{a}(t,q,\dot{q})\partial_{q^{a}}$
is the Lie generator, and the quantity

$\frac{\partial
I^{(m)}}{\partial\dot{q}^{i_{1}}}=M_{i_{1}}+2M_{i_{1}i_{2}}\dot{q}^{i_{2}}+3M_{i_{1}i_{2}i_{3}}\dot{q}^{i_{2}}\dot{q}^{i_{3}}+...+mM_{i_{1}i_{2}...i_{m}}\dot{q}^{i_{2}}...\dot{q}^{i_{m}}=\sum^{m-1}_{r=0}(r+1)M_{i_{1}i_{2}...i_{r+1}}\dot{q}^{i_{2}}...\dot{q}^{i_{r+1}}.$

##  4 FIs of order $m=1$: LFIs

Applying Theorem 1 for $m=1$ (LFIs), we find the following proposition for the
LFIs.

###### Proposition 3

There are two types of LFIs for the autonomous (in general non-Riemannian)
dynamical system (1) which are the following:

LFI 1.

$I^{(1)}_{n}=\sum^{n}_{N=0}L_{(N)a}t^{N}\dot{q}^{a}+s_{0}\frac{t^{n+1}}{n+1}+\sum^{n>0}_{N=1}L_{(N-1)a}Q^{a}\frac{t^{N}}{N}+G(q)$
(20)

where $n\geq 0$, $G(q)$ is an arbitrary smooth function, $L_{(N)a}(q)$ with
$N=0,1,...,n$ are generalized KVs satisfying the conditions:

$\displaystyle L_{(1)a}(n>0)$ $\displaystyle=$ $\displaystyle-G_{,a}.$ (21)
$\displaystyle\left(L_{(k-1)b}Q^{b}\right)_{,a}$ $\displaystyle=$
$\displaystyle-k(k+1)L_{(k+1)a}(k<n),\enskip k=1,2,...,n,\enskip n>0,$ (22)

and the constant $s_{0}$ is defined by the condition

$L_{(n)a}Q^{a}=s_{0}.$ (23)

LFI 2.

$I^{(1)}_{e}=\frac{e^{\lambda t}}{\lambda}\left(\lambda
L_{a}\dot{q}^{a}+L_{a}Q^{a}\right)$ (24)

where $\lambda\neq 0$ and $L_{a}(q)$ is a generalized KV satisfying the
condition

$\left(L_{b}Q^{b}\right)_{,a}=-\lambda^{2}L_{a}.$ (25)

The LFI (20) consists of two independent LFIs: a) A LFI for the even vectors
$L_{(2N)a}$, and b) another one for the odd vectors $L_{(2N+1)a}$ and the
scalar $G(q)$. This results directly from the fact that condition (21)
involves only the odd vector $L_{(1)a}$ and the scalar $G(q)$, while
conditions (22) are separated in two sets; one set involving only the even
vectors $L_{(2N)a}$ (i.e. for the odd values $k=1,3,5,...$), and another one
involving only the odd vectors $L_{(2N+1)a}$ (i.e. for the even values
$k=2,4,6,...$). We note that the above holds for either an odd or an even
time-dependence $n$. In the following proposition, we give the explicit
formulae of these two independent LFIs.

###### Proposition 4

The LFI $I^{(1)}_{n}$ given in (20) consists of the following two independent
LFIs:

LFI 1.1.

$I^{(1,1)}_{\ell}=\sum_{N=1}^{\ell}t^{2N-1}L_{(2N-1)a}\dot{q}^{a}+s_{1}\frac{t^{2\ell}}{2\ell}+\sum_{N=1}^{\ell-1\geq
1}L_{(2N-1)a}Q^{a}\frac{t^{2N}}{2N}+G(q)$ (26)

where $\ell>0$, $L_{(2N-1)a}(q)$ with $N=1,2,...,\ell$ are generalized KVs
satisfying the condition

$\left(L_{(2k-1)b}Q^{b}\right)_{,a}=-2k(2k+1)L_{(2k+1)a},\enskip
k=1,2,...,\ell-1,\enskip\ell>1,$ (27)

$G(q)$ is an arbitrary smooth function such that $L_{(1)a}=-G_{,a}$ is a
gradient generalized KV, and the constant $s_{1}$ is defined by the condition

$s_{1}=L_{(2\ell-1)a}Q^{a}.$ (28)

LFI 1.2.

$I^{(1,2)}_{\ell}=\sum_{N=0}^{\ell}t^{2N}L_{(2N)a}\dot{q}^{a}+s_{0}\frac{t^{2\ell+1}}{2\ell+1}+\sum_{N=0}^{\ell-1\geq
0}L_{(2N)a}Q^{a}\frac{t^{2N+1}}{2N+1}$ (29)

where $\ell\geq 0$, $L_{(2N)a}(q)$ with $N=0,1,...,\ell$ are generalized KVs
satisfying the condition

$\left(L_{(2k)b}Q^{b}\right)_{,a}=-2(2k+1)(k+1)L_{(2k+2)a},\enskip
k=0,1,...,\ell-1,\enskip\ell>0,$ (30)

and the constant $s_{0}$ is defined by the condition

$L_{(2\ell)a}Q^{a}=s_{0}.$ (31)

Therefore, the general autonomous dynamical system (1) admits three
independent LFIs which are given by the formulae (24), (26), and (29).

The notation $I^{(1,\alpha)}_{\ell}$, where $\alpha=1,2$, indicates one of the
two types of independent LFIs of the type $I^{(1)}_{n}$ with time-dependence
fixed by the lower index $\ell$.

We note that condition (27) is derived from (22) for the even values
$k=2,4,6,...,2\ell$ if we set $n=2\ell$ and rename the index $k$ as $2k$;
while condition (30) is derived from (22) for the odd values
$k=1,3,5,...,2\ell-1$ if we set $n=2\ell$ and rename the index $k$ as $2k+1$.

##  5 FIs of order $m>1$

Applying Theorem 1 for $m>1$, we find the following proposition.

###### Proposition 5

There are two types of $m$th-order FIs with $m>1$ for the autonomous (in
general non-Riemannian) dynamical system (1) which are the following:

Integral 1.

$I^{(m>1)}_{n}=\sum_{r=1}^{m}\left(\sum^{n}_{N=0}L_{(N)i_{1}...i_{r}}t^{N}\right)\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+s_{0}\frac{t^{n+1}}{n+1}+\sum^{n>0}_{N=1}L_{(N-1)c}Q^{c}\frac{t^{N}}{N}+G(q)$
(32)

where $m>1$, $n\geq 0$, $L_{(N)i_{1}...i_{m}}(q)$ with $N=0,1,...,n$ are
$\mathbf{m}$th-order generalized KTs satisfying the condition

$L_{(k)i_{1}...i_{m}}=-\frac{1}{k}L_{(k-1)(i_{1}...i_{m-1}|i_{m})},\enskip
k=1,2,...,n,\enskip n>0,$ (33)

the totally symmetric tensor $L_{(n)i_{1}...i_{m-1}}(q)$ is an
$\mathbf{(m-1)}$th-order generalized KT, the constant $s_{0}$ is defined by
the condition

$L_{(n)a}Q^{a}=s_{0}$ (34)

while the function $G(q)$ and the remaining totally symmetric tensors
$L_{(N)i_{1}...i_{r}}(q)$ satisfy the conditions:

$\displaystyle G_{,i_{1}}$ $\displaystyle=$ $\displaystyle
2L_{(0)i_{1}i_{2}}Q^{i_{2}}-L_{(1)i_{1}}(n>0)$ (35)
$\displaystyle\left(L_{(k-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle 2kL_{(k)i_{1}i_{2}}Q^{i_{2}}-k(k+1)L_{(k+1)i_{1}}(k<n),\enskip
k=1,2,...,n,\enskip n>0$ (36) $\displaystyle L_{(k)(i_{1}...i_{r-1}|i_{r})}$
$\displaystyle=$
$\displaystyle(r+1)L_{(k)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(k+1)L_{(k+1)i_{1}...i_{r}}(k<n),$
(37) $\displaystyle k=0,1,...,n,\enskip r=2,3,...,m-1,\enskip m>2.$

Integral 2.

$I^{(m>1)}_{e}=\frac{e^{\lambda
t}}{\lambda}\left(\lambda\sum^{m}_{r=1}L_{i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+L_{c}Q^{c}\right)$
(38)

where $\lambda\neq 0$, $L_{i_{1}...i_{m}}(q)$ is an $\mathbf{m}$th-order
generalized KT satisfying the condition

$L_{i_{1}...i_{m}}=-\frac{1}{\lambda}L_{(i_{1}...i_{m-1}|i_{m})}$ (39)

and the remaining totally symmetric tensors $L_{i_{1}...i_{r}}(q)$ with
$r=1,2,...,m-1$ satisfy the conditions:

$\displaystyle\left(L_{c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle 2\lambda L_{i_{1}i_{2}}Q^{i_{2}}-\lambda^{2}L_{i_{1}}$ (40)
$\displaystyle L_{(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-\lambda
L_{i_{1}...i_{r}},\enskip r=2,3,...,m-1,\enskip m>2.$ (41)

From Proposition 5, we observe that the conditions (33) - (37) of the $m$th-
order FI (32) are divided in two classes, according to if the tensor
quantities $L_{(N)i_{1}...i_{r}}(q)$ are of even or odd order, and the
associated index $N$ is even or odd. Therefore, the FI (32) consists of two
independent FIs:
a) One FI, say $I^{(m,1)}_{\ell}$, which contains tensor quantities
$L_{(2N-1)i_{1}...i_{r}}(q)$ of odd order and $L_{(2N)i_{1}...i_{r}}(q)$ of
even order.
b) Another FI, say $I^{(m,2)}_{\ell}$, which contains tensor quantities
$L_{(2N-1)i_{1}...i_{r}}(q)$ of even order and $L_{(2N)i_{1}...i_{r}}(q)$ of
odd order.

We note that the order $r$ of the involved totally symmetric tensors
$L_{(N)i_{1}...i_{r}}(q)$ takes values from $1$ to $m>1$, where $m$ is the
order of the considered FI (32).

The independent FIs $I^{(m,1)}_{\ell}$ and $I^{(m,2)}_{\ell}$ are given by the
following explicit formulae:

a.

$\displaystyle I^{(m>1,1)}_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{r=1,\text{odd}}^{m}\sum_{N=1}^{\ell>0}t^{2N-1}L_{(2N-1)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+\sum_{r=1,\text{even}}^{m}\sum_{N=0}^{\ell}t^{2N}L_{(2N)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+$
(42) $\displaystyle+\sum_{N=1}^{\ell>0}L_{(2N-1)c}Q^{c}\frac{t^{2N}}{2N}+G(q)$

where the time-dependence $\ell\geq 0$ and the involved quantities satisfy the
conditions:

$\displaystyle G_{,i_{1}}$ $\displaystyle=$ $\displaystyle
2L_{(0)i_{1}i_{2}}Q^{i_{2}}-L_{(1)i_{1}}(\ell>0)$ (43)
$\displaystyle\left(L_{(2N-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle
4NL_{(2N)i_{1}i_{2}}Q^{i_{2}}-2N(2N+1)L_{(2N+1)i_{1}}(N<\ell),\enskip
N=1,2,...,\ell,\enskip\ell>0$ (44) $\displaystyle
L_{(2N-1)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N-1)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-2NL_{(2N)i_{1}...i_{r}},$
(45) $\displaystyle N=1,2,...,\ell,\enskip r=2,4,6,...,\enskip\ell>0$
$\displaystyle L_{(2N)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(2N+1)L_{(2N+1)i_{1}...i_{r}}(N<\ell),$
(46) $\displaystyle N=0,1,...,\ell,\enskip r=3,5,7,...~{}.$

In conditions (45) and (46), it holds that $r\leq m-1$ where $m>2$.

b.

$\displaystyle I^{(m>1,2)}_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{r=1,\text{odd}}^{m}\sum_{N=0}^{\ell}t^{2N}L_{(2N)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+\sum_{r=1,\text{even}}^{m}\sum_{N=1}^{\ell>0}t^{2N-1}L_{(2N-1)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+$
(47) $\displaystyle+s_{0}\frac{t^{2\ell+1}}{2\ell+1}+\sum_{N=0}^{\ell-1\geq
0}L_{(2N)c}Q^{c}\frac{t^{2N+1}}{2N+1}$

where the time-dependence $\ell\geq 0$ and the involved quantities satisfy the
conditions:

$\displaystyle L_{(2\ell)a}Q^{a}$ $\displaystyle=$ $\displaystyle s_{0}$ (48)
$\displaystyle\left(L_{(2N-2)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle
2(2N-1)L_{(2N-1)i_{1}i_{2}}Q^{i_{2}}-2N(2N-1)L_{(2N)i_{1}},\enskip
N=1,2,...,\ell,\enskip\ell>0$ (49) $\displaystyle
L_{(2N)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(2N+1)L_{(2N+1)i_{1}...i_{r}}(N<\ell),$
(50) $\displaystyle N=0,1,...,\ell,\enskip r=2,4,6,...,$ $\displaystyle
L_{(2N-1)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N-1)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-2NL_{(2N)i_{1}...i_{r}},$
(51) $\displaystyle N=1,2,...,\ell,\enskip r=3,5,7,...,\enskip\ell>0.$

In conditions (50) and (51), it holds that $r\leq m-1$ where $m>2$.

For the FIs (42) and (47), we have also that $L_{(2\ell)i_{1}...i_{m-1}}(q)$
is an $\mathbf{(m-1)}$th-order generalized KT, and $L_{(N)i_{1}...i_{m}}(q)$
with $N=0,1,...,2\ell$ are $\mathbf{m}$th-order generalized KTs such that:

$\displaystyle L_{(2N)i_{1}...i_{m}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2N}L_{(2N-1)(i_{1}...i_{m-1}|i_{m})},\enskip
N=1,2,...,\ell,\enskip\ell>0$ (52) $\displaystyle L_{(2N-1)i_{1}...i_{m}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2N-1}L_{(2N-2)(i_{1}...i_{m-1}|i_{m})},\enskip
N=1,2,...,\ell,\enskip\ell>0.$ (53)

We note that the sum $\sum_{r=1,\text{odd}}^{m}$ is over the odd values of
$r$, while the sum $\sum_{r=1,\text{even}}^{m}$ is over the even values of
$r$.

##  6 How to remove the $(m-1)$th-order geometric symmetries from the two
independent FIs $I^{(m>1,1)}_{\ell}$ and $I^{(m>1,2)}_{\ell}$: The complete
forms

As we have seen, the two independent FIs (42) and (47) include geometric
symmetries of order smaller from the order $m>1$ of the FIs. These symmetries
are described by the $(m-1)$th-order generalized KT
$L_{(2\ell)i_{1}...i_{m-1}}(q)$. We observe that if $m=2\nu$ (even) where
$\nu>0$, the considered $(m-1)$th-order generalized KT appears in the FI (47);
while in the case that $m=2\nu+1$ (odd), appears in the FI (42). Moreover, for
an even order $m$, the condition (52) accompanies the FI (42) and the
condition (53) accompanies the FI (47); while for an odd order $m$, the
condition (52) accompanies the FI (47) and the condition (53) accompanies the
FI (42).

The $(m-1)$th-order generalized KT $L_{(2\ell)i_{1}...i_{m-1}}(q)$ can be
removed from the expressions (42) and (47) by applying the following
procedure:

1) For an even order $m=2\nu$ where $\nu>0$.

In this case, the $(2\nu-1)$th-order generalized KT
$L_{(2\ell)i_{1}...i_{2\nu-1}}(q)$ appears in the independent FI (47). We can
remove this $(2\nu-1)$th-order symmetry by introducing a sequence of an even
rank totally symmetric tensors of a higher time-dependence, that is, we add in
the FI (47) terms of the form

$t^{2\ell+1}L_{(2\ell+1)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}},\enskip
r=2,4,...,2\nu,\enskip\nu>0$ (54)

such that $L_{(2\ell+1)i_{1}...i_{2\nu}}$ is a $(2\nu)$th-order generalized KT
given by the relation

$L_{(2\ell+1)i_{1}...i_{2\nu}}=-\frac{1}{2\ell+1}L_{(2\ell)(i_{1}...i_{2\nu-1}|i_{2\nu})}.$
(55)

Then, the condition (48) must be generalized as

$\left(L_{(2\ell)c}Q^{c}\right)_{,i_{1}}=2(2\ell+1)L_{(2\ell+1)i_{1}i_{2}}Q^{i_{2}};$
(56)

from the condition (50), the restriction $N<\ell$ must be removed because now
quantities of the form $L_{(2N+1)i_{1}...i_{r}}$ do exist for $N=\ell$; and we
must add the condition

$L_{(2\ell+1)(i_{1}...i_{r-1}|i_{r})}=(r+1)L_{(2\ell+1)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}},\enskip
r=3,5,7,...,2\nu-1\enskip\nu>1.$ (57)

Observe that the condition (55) removes the original $(2\nu-1)$th-order
symmetry by allowing the $(2\nu-1)$th-order generalized KT
$L_{(2\ell)i_{1}...i_{2\nu-1}}(q)$ to be a totally symmetric tensor of the
same rank, which is not necessarily a KT. Then, the case of the generalized KT
is derived as a subcase. This shows that symmetries of an order lesser than
the order of the FI can always be absorbed into a higher order term.

2) For an odd order $m=2\nu+1$ where $\nu>0$.

In this case, the $(2\nu)$th-order generalized KT
$L_{(2\ell)i_{1}...i_{2\nu}}(q)$ appears in the independent FI (42). We can
remove this $(2\nu)$th-order symmetry by introducing a sequence of an odd rank
totally symmetric tensors of a higher time-dependence, that is, we add in the
FI (42) the sum

$\sum_{r=1,odd}^{2\nu+1}t^{2\ell+1}L_{(2\ell+1)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+s_{1}\frac{t^{2\ell+2}}{2\ell+2}$
(58)

such that $L_{(2\ell+1)i_{1}...i_{2\nu+1}}$ is a $(2\nu+1)$th-order
generalized KT given by the relation

$L_{(2\ell+1)i_{1}...i_{2\nu+1}}=-\frac{1}{2\ell+1}L_{(2\ell)(i_{1}...i_{2\nu}|i_{2\nu+1})}$
(59)

and the constant $s_{1}$ is defined by the relation

$L_{(2\ell+1)c}Q^{c}=s_{1}.$ (60)

Then, from conditions (44) and (46), the restriction $N<\ell$ must be removed
because now quantities of the form $L_{(2N+1)i_{1}...i_{r}}$ do exist for
$N=\ell$; and we must add the condition

$L_{(2\ell+1)(i_{1}...i_{r-1}|i_{r})}=(r+1)L_{(2\ell+1)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}},\enskip
r=2,4,6,...,2\nu.$ (61)

Observe that the condition (59) removes the original $(2\nu)$th-order symmetry
defined by allowing the $(2\nu)$th-order generalized KT
$L_{(2\ell)i_{1}...i_{2\nu}}(q)$ to be a totally symmetric tensor of the same
rank, which is not necessarily a KT. Then, the case of the generalized KT is
derived as a subcase. This shows that symmetries of an order lesser than the
order of the FI can always be absorbed into a higher order term.

We collect the above results in the following propositions.

###### Proposition 6

For an even order $m=2\nu$, where $\nu>0$, the complete forms (i.e.
expressions without geometric symmetries of order less than $m$) of the
independent FIs (42) and (47) are the following:

Integral 1.1.

$\displaystyle J^{(2\nu,1)}_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{r=1,\text{odd}}^{2\nu}\sum_{N=1}^{\ell>0}t^{2N-1}L_{(2N-1)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+\sum_{r=1,\text{even}}^{2\nu}\sum_{N=0}^{\ell}t^{2N}L_{(2N)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+$
(62) $\displaystyle+\sum_{N=1}^{\ell>0}L_{(2N-1)c}Q^{c}\frac{t^{2N}}{2N}+G(q)$

where $\ell\geq 0$, $L_{(2N)i_{1}...i_{2\nu}}(q)$ with $N=0,1,...,\ell$ are
$\mathbf{(2\nu)}$th-order generalized KTs given by the relation

$L_{(2N)i_{1}...i_{2\nu}}=-\frac{1}{2N}L_{(2N-1)(i_{1}...i_{2\nu-1}|i_{2\nu})},\enskip
N=1,2,...,\ell,\enskip\ell>0$ (63)

and the involved quantities satisfy the conditions:

$\displaystyle G_{,i_{1}}$ $\displaystyle=$ $\displaystyle
2L_{(0)i_{1}i_{2}}Q^{i_{2}}-L_{(1)i_{1}}(\ell>0)$ (64)
$\displaystyle\left(L_{(2N-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle
4NL_{(2N)i_{1}i_{2}}Q^{i_{2}}-2N(2N+1)L_{(2N+1)i_{1}}(N<\ell),\enskip
N=1,2,...,\ell,\enskip\ell>0$ (65) $\displaystyle
L_{(2N-1)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N-1)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-2NL_{(2N)i_{1}...i_{r}},$
(66) $\displaystyle N=1,2,...,\ell,\enskip
r=2,4,6,...,2\nu-2,\enskip\ell>0,\enskip\nu>1$ $\displaystyle
L_{(2N)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(2N+1)L_{(2N+1)i_{1}...i_{r}}(N<\ell),$
(67) $\displaystyle N=0,1,...,\ell,\enskip r=3,5,7,...,2\nu-1,\enskip\nu>1.$

Integral 1.2.

$\displaystyle J^{(2\nu,2)}_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{r=1,\text{odd}}^{2\nu}\sum_{N=0}^{\ell}t^{2N}L_{(2N)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+\sum_{r=1,\text{even}}^{2\nu}\sum_{N=1}^{\ell+1}t^{2N-1}L_{(2N-1)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+$
(68) $\displaystyle+s_{0}\frac{t^{2\ell+1}}{2\ell+1}+\sum_{N=0}^{\ell-1\geq
0}L_{(2N)c}Q^{c}\frac{t^{2N+1}}{2N+1}$

where $\ell\geq 0$, $L_{(2N-1)i_{1}...i_{2\nu}}(q)$ with $N=1,2,...,\ell+1$
are $\mathbf{(2\nu)}$th-order generalized KTs given by the relation

$L_{(2N-1)i_{1}...i_{2\nu}}=-\frac{1}{2N-1}L_{(2N-2)(i_{1}...i_{2\nu-1}|i_{2\nu})},\enskip
N=1,2,...,\ell+1$ (69)

and the involved quantities satisfy the conditions:

$\displaystyle\left(L_{(2N-2)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle
2(2N-1)L_{(2N-1)i_{1}i_{2}}Q^{i_{2}}-2N(2N-1)L_{(2N)i_{1}}(N<\ell+1),$ (70)
$\displaystyle N=1,2,...,\ell+1$ $\displaystyle
L_{(2N)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(2N+1)L_{(2N+1)i_{1}...i_{r}},$
(71) $\displaystyle N=0,1,...,\ell,\enskip r=2,4,6,...,2\nu-2,\enskip\nu>1$
$\displaystyle L_{(2N-1)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N-1)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-2NL_{(2N)i_{1}...i_{r}}(N<\ell+1),$
(72) $\displaystyle N=1,2,...,\ell+1,\enskip r=3,5,7,...,2\nu-1,\enskip\nu>1.$

###### Proposition 7

For an odd order $m=2\nu+1$, where $\nu>0$, the complete forms (i.e.
expressions without geometric symmetries of order less than $m$) of the
independent FIs (42) and (47) are the following:

Integral 1.1.

$\displaystyle J^{(2\nu+1,1)}_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{r=1,\text{odd}}^{2\nu+1}\sum_{N=1}^{\ell+1}t^{2N-1}L_{(2N-1)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+\sum_{r=1,\text{even}}^{2\nu+1}\sum_{N=0}^{\ell}t^{2N}L_{(2N)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+$
(73)
$\displaystyle+s_{1}\frac{t^{2\ell+2}}{2\ell+2}+\sum_{N=1}^{\ell>0}L_{(2N-1)c}Q^{c}\frac{t^{2N}}{2N}+G(q)$

where $\ell\geq 0$, $L_{(2N-1)i_{1}...i_{2\nu+1}}(q)$ with $N=1,...,\ell+1$
are $\mathbf{(2\nu+1)}$th-order generalized KTs given by the relation

$L_{(2N-1)i_{1}...i_{2\nu+1}}=-\frac{1}{2N-1}L_{(2N-2)(i_{1}...i_{2\nu}|i_{2\nu+1})},\enskip
N=1,2,...,\ell+1$ (74)

while $s_{1}$ is a constant defined by the relation

$L_{(2\ell+1)a}Q^{a}=s_{1}$ (75)

and the involved quantities satisfy the conditions:

$\displaystyle G_{,i_{1}}$ $\displaystyle=$ $\displaystyle
2L_{(0)i_{1}i_{2}}Q^{i_{2}}-L_{(1)i_{1}}(\ell>0)$ (76)
$\displaystyle\left(L_{(2N-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle 4NL_{(2N)i_{1}i_{2}}Q^{i_{2}}-2N(2N+1)L_{(2N+1)i_{1}},\enskip
N=1,2,...,\ell,\enskip\ell>0$ (77) $\displaystyle
L_{(2N-1)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N-1)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-2NL_{(2N)i_{1}...i_{r}}(N<\ell+1),$
(78) $\displaystyle N=1,2,...,\ell+1,\enskip r=2,4,6,...,2\nu$ $\displaystyle
L_{(2N)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(2N+1)L_{(2N+1)i_{1}...i_{r}},$
(79) $\displaystyle N=0,1,...,\ell,\enskip r=3,5,7,...,2\nu-1,\enskip\nu>1.$

Integral 1.2.

$\displaystyle J^{(2\nu+1,2)}_{\ell}$ $\displaystyle=$
$\displaystyle\sum_{r=1,\text{odd}}^{2\nu+1}\sum_{N=0}^{\ell}t^{2N}L_{(2N)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+\sum_{r=1,\text{even}}^{2\nu+1}\sum_{N=1}^{\ell>0}t^{2N-1}L_{(2N-1)i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+$
(80) $\displaystyle+s_{0}\frac{t^{2\ell+1}}{2\ell+1}+\sum_{N=0}^{\ell-1\geq
0}L_{(2N)c}Q^{c}\frac{t^{2N+1}}{2N+1}$

where $\ell\geq 0$, $L_{(2N)i_{1}...i_{2\nu+1}}(q)$ with $N=0,1,...,\ell$ are
$\mathbf{(2\nu+1)}$th-order generalized KTs given by the relation

$L_{(2N)i_{1}...i_{2\nu+1}}=-\frac{1}{2N}L_{(2N-1)(i_{1}...i_{2\nu}|i_{2\nu+1})},\enskip
N=1,2,...,\ell,\enskip\ell>0$ (81)

and the involved quantities satisfy the conditions:

$\displaystyle L_{(2\ell)a}Q^{a}$ $\displaystyle=$ $\displaystyle s_{0}$ (82)
$\displaystyle\left(L_{(2N-2)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle
2(2N-1)L_{(2N-1)i_{1}i_{2}}Q^{i_{2}}-2N(2N-1)L_{(2N)i_{1}},\enskip
N=1,2,...,\ell,\enskip\ell>0$ (83) $\displaystyle
L_{(2N)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(2N+1)L_{(2N+1)i_{1}...i_{r}}(N<\ell),$
(84) $\displaystyle N=0,1,...,\ell,\enskip r=2,4,6,...,2\nu$ $\displaystyle
L_{(2N-1)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(2N-1)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-2NL_{(2N)i_{1}...i_{r}},$
(85) $\displaystyle N=1,2,...,\ell,\enskip
r=3,5,7,...,2\nu-1,\enskip\ell>0,\enskip\nu>1.$

Therefore, concerning the higher order FIs of a general autonomous dynamical
system, we have the following general result.

###### Proposition 8

The autonomous (non-Riemannian in general) dynamical system (1) admits three
independent $m$th-order FIs –autonomous or time-dependent– with $m>0$, which
are given by the following formulae:
i. For $m=1$, we have the LFIs (24), (26) and (29).
ii. For $m=2\nu$ with $\nu>0$, we have the FIs (38), (62) and (68).
iii. For $m=2\nu+1$ with $\nu>0$, we have the FIs (38), (73) and (80).

In the following sections, we give applications of the above general results.

##  7 Application 1: A family of 2d non-Riemannian autonomous dynamical
systems

We consider two-dimensional (2d) autonomous dynamical systems of the form:

$\displaystyle\ddot{x}$ $\displaystyle=$
$\displaystyle-Q^{1}(x,y)-\Gamma^{1}_{11}(x,y)\dot{x}^{2}$ (86)
$\displaystyle\ddot{y}$ $\displaystyle=$
$\displaystyle-Q^{2}(x,y)-\Gamma^{2}_{22}(x,y)\dot{y}^{2}$ (87)

where $q^{a}=(x,y)$ are the generalized coordinates, $Q^{1},Q^{2}$ are the
components of the generalized forces, and $\Gamma^{1}_{11},\Gamma^{2}_{22}$
are the non-zero symmetric connection coefficients of the system (all the
other connection coefficients vanish).

We will find conditions for which the dynamical system (86) - (87) is non-
Riemannian. In that case, a kinetic metric (i.e. a regular Lagrangian or a
kinetic energy) cannot be defined, and standard methods (e.g. Noether’s
theorem) for the determination of FIs cannot be applied. Therefore, Theorem 1
is the only systematic method we have in order to determine the FIs of the
system (86) - (87).

###  7.1 Conditions for a non-Riemannian connection

It is well-known that the symmetric connection coefficients $\Gamma^{a}_{bc}$
define a Riemannian connection iff there exists a (kinetic) metric
$\gamma_{ab}$ with zero metricity, that is, $\gamma_{ab|c}=0$. Then,

$\Gamma^{a}_{bc}=\frac{1}{2}\gamma^{ad}\left(\gamma_{bd,c}+\gamma_{cd,b}-\gamma_{bc,d}\right).$

Question: _In which cases the dynamical system ( 86) - (87) is defined over a
Riemannian configuration space or, equivalently, when the associated symmetric
connection is Riemannian?_

In order to answer this question, we assume that there exists a kinetic metric
$\gamma_{ab}(x,y)$ such that

$\gamma_{ab|c}=0\implies\gamma_{ab,c}-\gamma_{db}\Gamma^{d}_{ac}-\gamma_{ad}\Gamma^{d}_{bc}=0.$
(88)

Setting $\Gamma^{1}_{11}\equiv f_{1}(x,y)$ and $\Gamma^{2}_{22}\equiv
f_{2}(x,y)$, condition (88) gives the following system of PDEs:

$\begin{cases}\gamma_{11,x}=2\gamma_{11}f_{1},\enskip\gamma_{12,x}=\gamma_{12}f_{1},\enskip\gamma_{22,x}=0\implies\gamma_{22}=\gamma_{22}(y)\\\
\gamma_{22,y}=2\gamma_{22}f_{2},\enskip\gamma_{12,y}=\gamma_{12}f_{2},\enskip\gamma_{11,y}=0\implies\gamma_{11}=\gamma_{11}(x)\end{cases}\implies$
$\displaystyle\frac{d\gamma_{11}}{dx}$ $\displaystyle=$ $\displaystyle
2\gamma_{11}(x)f_{1}$ (89) $\displaystyle\frac{d\gamma_{22}}{dy}$
$\displaystyle=$ $\displaystyle 2\gamma_{22}(y)f_{2}$ (90)
$\displaystyle\gamma_{12,x}$ $\displaystyle=$
$\displaystyle\gamma_{12}(x,y)f_{1}$ (91) $\displaystyle\gamma_{12,y}$
$\displaystyle=$ $\displaystyle\gamma_{12}(x,y)f_{2}.$ (92)

In order to have a well-defined metric $\gamma_{ab}$ (i.e. the inverse
$\gamma^{ab}$ exists), it must also hold that

$\det\left[\gamma_{ab}\right]\neq
0\implies\gamma_{11}\gamma_{22}-\gamma_{12}^{2}\neq 0.$ (93)

We consider the following cases:

1) Case $\gamma_{12}\neq 0$.

1.1. Subcase $\gamma_{11}=\gamma_{22}=0$.

Conditions (89), (90) and (93) are satisfied identically.

The remaining conditions (91) and (92) give the Riemannian connection
coefficients:

$f_{1}=\frac{F_{,x}}{F},\enskip f_{2}=\frac{F_{,y}}{F}$

where $\gamma_{12}\equiv F(x,y)$.

The associated kinetic metric is
$\gamma_{ab}=F(x,y)\left(\begin{array}[]{cc}0&1\\\ 1&0\\\ \end{array}\right)$.
The constrained geodesics of this metric has been discussed for various cases
of the function $F(x,y)$ in [11] and, more recently, in [12].

1.2. Subcase $\gamma_{11}\equiv f(x)\neq 0$ and $\gamma_{22}=0$.

Conditions (90) and (93) are satisfied identically.

Condition (89) implies that $f_{1}=\frac{f_{,x}}{2f}$. Replacing $f_{1}$ in
(91), we find that $\gamma_{12}=h(y)\sqrt{f(x)}$ which when replaced into the
remaining condition (92) gives $f_{2}=\frac{h_{,y}}{h}$.

The associated kinetic metric is
$\gamma_{ab}=\left(\begin{array}[]{cc}f(x)&h(y)\sqrt{f}\\\ h(y)\sqrt{f}&0\\\
\end{array}\right)$.

1.3. Subcase $\gamma_{22}\equiv h(y)\neq 0$ and $\gamma_{11}=0$.

Conditions (89) and (93) are satisfied identically.

Condition (90) implies that $f_{2}=\frac{h_{,y}}{2h}$. Replacing $f_{2}$ in
(92), we find that $\gamma_{12}=f(x)\sqrt{h(y)}$ which when replaced into the
remaining condition (91) gives $f_{1}=\frac{f_{,x}}{f}$.

The associated kinetic metric is
$\gamma_{ab}=\left(\begin{array}[]{cc}0&f(x)\sqrt{h}\\\ f(x)\sqrt{h}&h(y)\\\
\end{array}\right)$.

1.4. Subcase $\gamma_{11}\equiv f(x),\gamma_{22}\equiv h(y)$, and
$f(x)h(y)\neq 0$.

Conditions (89) and (90) imply that $f_{1}=\frac{f_{,x}}{2f}$ and
$f_{2}=\frac{h_{,y}}{2h}$, respectively. Replacing $f_{1},f_{2}$ in the
remaining conditions (91), (92), we find that
$\gamma_{12}=c_{0}\sqrt{f(x)h(y)}$ where $c_{0}$ is an arbitrary non-zero
constant.

The associated kinetic metric is
$\gamma_{ab}=\left(\begin{array}[]{cc}f(x)&c_{0}\sqrt{fh}\\\
c_{0}\sqrt{fh}&h(y)\\\ \end{array}\right)$.

From the condition (93), we find that $c_{0}\neq\pm 1$.

2) Case $\gamma_{12}=0$.

Condition (93) implies that $\gamma_{11}\gamma_{22}\neq 0$.

We set $\gamma_{11}\equiv f(x)$ and $\gamma_{22}\equiv h(y)$.

Conditions (91) and (92) are satisfied identically.

The remaining conditions (89) and (90) give $f_{1}=\frac{f_{,x}}{2f}$ and
$f_{2}=\frac{h_{,y}}{2h}$, respectively.

The associated kinetic metric is
$\gamma_{ab}=\left(\begin{array}[]{cc}f(x)&0\\\ 0&h(y)\\\ \end{array}\right)$.

We collect the above results in the following Proposition.

###### Proposition 9

Autonomous dynamical systems of the form (86) - (87) admit a Riemannian
geometry (i.e. the quantities $\Gamma^{a}_{bc}$ are Riemannian connection
coefficients defined by a kinetic metric $\gamma_{ab}$) in the following
cases:
1) For $\Gamma^{1}_{11}=\frac{F_{,x}}{F}$ and
$\Gamma^{2}_{22}=\frac{F_{,y}}{F}$, where $F(x,y)$ is a non-zero arbitrary
smooth function. The associated kinetic metric is
$\gamma_{ab}=F(x,y)\left(\begin{array}[]{cc}0&1\\\ 1&0\\\ \end{array}\right)$.
2) For $\Gamma^{1}_{11}=\frac{f_{,x}}{2f}$ and
$\Gamma^{2}_{22}=\frac{h_{y}}{h}$, where $f(x)$ and $h(y)$ are non-zero
arbitrary smooth functions. Then,
$\gamma_{ab}=\left(\begin{array}[]{cc}f(x)&h(y)\sqrt{f}\\\ h(y)\sqrt{f}&0\\\
\end{array}\right)$.
3) For $\Gamma^{1}_{11}=\frac{f_{x}}{f}$ and
$\Gamma^{2}_{22}=\frac{h_{,y}}{2h}$. Then,
$\gamma_{ab}=\left(\begin{array}[]{cc}0&f(x)\sqrt{h}\\\ f(x)\sqrt{h}&h(y)\\\
\end{array}\right)$.
4) For $\Gamma^{1}_{11}=\frac{f_{,x}}{2f}$ and
$\Gamma^{2}_{22}=\frac{h_{,y}}{2h}$. Then,
$\gamma_{ab}=\left(\begin{array}[]{cc}f(x)&c_{0}\sqrt{fh}\\\
c_{0}\sqrt{fh}&h(y)\\\ \end{array}\right)$ where the (possibly zero) constant
$c_{0}\neq\pm 1$.

From Proposition 9, we find the following proposition.

###### Proposition 10

The dynamical system (86) - (87) is non-Riemannian if the non-zero symmetric
connection coefficients $\Gamma^{1}_{11}(x,y)$ and $\Gamma^{2}_{22}(x,y)$
satisfy the condition

$\Gamma^{1}_{11,y}\neq\Gamma^{2}_{22,x}.$ (94)

Proof:
\- Case 1) of Proposition 9 implies that $\Gamma^{1}_{11}=(\ln F)_{,x}$ and
$\Gamma^{2}_{22}=(\ln F)_{,y}$. Taking the integrability condition $(\ln
F)_{,xy}=(\ln F)_{,yx}$, we find $\Gamma^{1}_{11,y}=\Gamma^{2}_{22,x}$.
Therefore, for a non-Riemannian connection, the condition (94) is required.
\- For the remaining cases 2) - 4) of Proposition 9, we observe that
$\Gamma^{1}_{11}(x)$ and $\Gamma^{2}_{22}(y)$. This implies that
$\Gamma^{1}_{11,y}=\Gamma^{2}_{22,x}=0$, which is a subcase of the condition
$\Gamma^{1}_{11,y}=\Gamma^{2}_{22,x}$. Therefore, for a non-Riemannian
connection, we find again the condition (94).

###  7.2 LFIs for the non-Riemannian dynamical system (86) - (87)

We assume that the connection of the autonomous dynamical system (86) - (87)
is non-Riemannian; therefore, the condition (94) applies.

Applying Theorem 1 for $m=1$ (LFIs) and $n=0$ (zero degree of time-
dependence), we find for the considered dynamical system the autonomous LFI

$I=L_{1}(x,y)\dot{x}+L_{2}(x,y)\dot{y}+s_{0}t$ (95)

where $L_{a}(x,y)$ is a generalized KV of the connection
$\Gamma^{a}_{bc}(x,y)$ defined by the dynamical equations (86) - (87) and
$s_{0}$ is a constant defined by the condition

$L_{1}Q^{1}+L_{2}Q^{2}=s_{0}.$ (96)

The generalized KV condition $L_{(a|b)}=0$ implies the following system of
PDEs:

$\displaystyle L_{1,x}-L_{1}\Gamma^{1}_{11}$ $\displaystyle=$ $\displaystyle
0$ (97) $\displaystyle L_{2,y}-L_{2}\Gamma^{2}_{22}$ $\displaystyle=$
$\displaystyle 0$ (98) $\displaystyle L_{1,y}+L_{2,x}$ $\displaystyle=$
$\displaystyle 0.$ (99)

We have an overdetermined system of four PDEs (96) - (99), which has six
unknown functions $L_{a}(x,y)$, $Q^{a}(x,y)$, $\Gamma^{1}_{11}(x,y)$,
$\Gamma^{2}_{22}(x,y)$ and one free parameter $s_{0}$. Therefore, in order to
solve it, we should fix either the dynamics of the system (i.e. the
generalized forces $Q^{a}$) or the non-Riemannian geometry of the system (i.e.
the non-Riemannian connection coefficients; it holds that
$\Gamma^{1}_{11,y}\neq\Gamma^{2}_{22,x}$).

####  7.2.1 Two linearly coupled harmonic oscillators with a non-Riemannian
quadratic damping term

As a first application, we fix the generalized forces $Q^{a}$. We assume that

$Q^{a}=\left(\begin{array}[]{c}kx-py\\\ ky+px\\\ \end{array}\right)$ (100)

where $k,p$ are arbitrary non-zero constants.

For the choice (100), the dynamical system (86) - (87) becomes:

$\displaystyle\ddot{x}$ $\displaystyle=$ $\displaystyle-
kx+py-\Gamma^{1}_{11}(x,y)\dot{x}^{2}$ (101) $\displaystyle\ddot{y}$
$\displaystyle=$ $\displaystyle-ky-px-\Gamma^{2}_{22}(x,y)\dot{y}^{2}$ (102)

which describes two linearly coupled harmonic oscillators with a non-
Riemannian (i.e. $\Gamma^{1}_{11,y}\neq\Gamma^{2}_{22,x}$) quadratic damping
term.

Replacing $Q^{a}$ from (100), equation (96) gives

$L_{1}=-\frac{ky+px}{kx-py}L_{2}+\frac{s_{0}}{kx-py}=\frac{1}{py-
kx}\left[(ky+px)L_{2}-s_{0}\right]$ (103)

which when substituted into (99) implies that

$L_{2}=(py-
kx)F_{1}\left(p(y^{2}-x^{2})-2kxy\right)-\frac{s_{0}x}{p(y^{2}-x^{2})-2kxy}$
(104)

where $F_{1}\left(p(y^{2}-x^{2})-2kxy\right)$ is an arbitrary smooth function
of its argument.

Replacing (104) in (103), we find

$L_{1}=(ky+px)F_{1}\left(p(y^{2}-x^{2})-2kxy\right)+\frac{s_{0}x(ky+px)}{(kx-
py)\left[p(y^{2}-x^{2})-2kxy\right]}+\frac{s_{0}}{kx-py}.$ (105)

Using (104) and (105), the remaining PDEs (97) and (98) determine the
connection coefficients as follows:

$\Gamma^{1}_{11}=\frac{L_{1,x}}{L_{1}},\enskip\Gamma^{2}_{22}=\frac{L_{2,y}}{L_{2}}.$
(106)

In order to have a non-Riemannian connection, the condition (94) must be
satisfied. Therefore,

$\Gamma^{1}_{11,y}\neq\Gamma^{2}_{22,x}\implies\left(\ln|L_{1}|\right)_{,xy}\neq\left(\ln|L_{2}|\right)_{,yx}\implies\left(\ln\left|\frac{L_{1}}{L_{2}}\right|\right)_{,xy}\neq
0$ (107)

which leads to restrictions among the parameters $k,p,s_{0}$ and the function
$F_{1}$.

We note that the vector $L_{a}$ whose components are given by the relations
(104) and (105) is a generalized KV of the connection (106).

Finally, replacing (106) in the dynamical equations (101) - (102), we find the
family of dynamical systems:

$\displaystyle\ddot{x}$ $\displaystyle=$ $\displaystyle-
kx+py-\frac{L_{1,x}}{L_{1}}\dot{x}^{2}$ (108) $\displaystyle\ddot{y}$
$\displaystyle=$ $\displaystyle-ky-px-\frac{L_{2,y}}{L_{2}}\dot{y}^{2}$ (109)

parameterized by the constant $s_{0}$ and the function $F_{1}$, which admits
the time-dependent LFI (95)

In order to get a specific example, we fix the parameters $s_{0}$ and $F_{1}$
as follows:

\- Case $s_{0}=0$ and $F_{1}=1$.

Then, equations (104) and (105) give the generalized KV

$L_{a}=\left(\begin{array}[]{c}ky+px\\\ py-kx\\\ \end{array}\right)$ (110)

which when replaced into the relations (106) determines the connection
coefficients:

$\Gamma^{1}_{11}=\frac{p}{ky+px},\enskip\Gamma^{2}_{22}=\frac{p}{py-kx}.$
(111)

In order to have a non-Riemannian connection, the quantities (111) must
satisfy the condition (107). We compute:

$\Gamma^{1}_{11,y}=-\frac{kp}{(ky+px)^{2}}\neq\Gamma^{2}_{22,x}=\frac{kp}{(py-
kx)^{2}}\implies$ $(py-kx)^{2}+(ky+px)^{2}\neq
0\implies(k^{2}+p^{2})(x^{2}+y^{2})\neq 0\implies k\neq\pm ip.$

Therefore, _the connection ( 111) is non-Riemannian only when $k\neq\pm ip$_.

Using (110), the dynamical system (108) - (109) becomes:

$\displaystyle\ddot{x}$ $\displaystyle=$ $\displaystyle-
kx+py-\frac{p}{ky+px}\dot{x}^{2}$ (112) $\displaystyle\ddot{y}$
$\displaystyle=$ $\displaystyle-ky-px-\frac{p}{py-kx}\dot{y}^{2}$ (113)

and the associated autonomous LFI (95) is

$I_{1}=(ky+px)\dot{x}+(py-kx)\dot{y}.$ (114)

Remark: In the case that $k=\pm ip$, the connection coefficients (111) reduce
to the Riemannian connection:

$\Gamma^{1}_{11\pm}=\frac{1}{x\pm iy},\enskip\Gamma^{2}_{22\pm}=\frac{1}{y\mp
ix}$

which, according to Proposition 9, is associated with a metric of the form 1).
Indeed, for an arbitrary function $F(x,y)$, we have:

$\begin{cases}(\ln F_{\pm})_{,x}=\frac{1}{x\pm iy}\\\ (\ln
F_{\pm})_{,y}=\frac{1}{y\mp ix}\end{cases}\implies
F_{+}=\frac{x^{2}+y^{2}}{y+ix},\enskip F_{-}=y+ix$

and the corresponding metric is
$\gamma_{ab}=F_{\pm}\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)$.

##  8 Application 2: The QFIs of a non-Riemannian dynamical system

Consider the dynamical system111This dynamical system has been suggested to us
by Dr. Andronikos Paliathanasis.:

$\displaystyle\ddot{u}$ $\displaystyle=$
$\displaystyle-\frac{8\beta}{u^{3}}\left(u\dot{u}\dot{w}-w\dot{u}^{2}\right)-\frac{1}{u^{2}}$
(115) $\displaystyle\ddot{w}$ $\displaystyle=$
$\displaystyle-\frac{4\beta}{u^{3}}\left(u\dot{w}^{2}-4w\dot{u}\dot{w}\right)+\frac{2w}{u^{3}}$
(116)

where $\beta$ is an arbitrary real constant. This system is autonomous
holonomic of the form (1) with variables

$q^{a}=\left(\begin{array}[]{c}u\\\ w\\\ \end{array}\right),\enskip
Q^{a}=\frac{1}{u^{2}}\left(\begin{array}[]{c}1\\\ -\frac{2w}{u}\\\
\end{array}\right).$

The symmetric connection coefficients are read from the dynamical equations
and are:

$\Gamma^{1}_{22}=\Gamma^{2}_{11}=0,\enskip\Gamma^{1}_{11}=\Gamma^{2}_{12}=-8\beta\frac{w}{u^{3}},\enskip\Gamma^{1}_{12}=\Gamma^{2}_{22}=\frac{4\beta}{u^{2}}.$
(117)

The non-zero components of the curvature tensor
$R^{a}{}_{bcd}=\Gamma^{a}_{bd,c}-\Gamma^{a}_{bc,d}+\Gamma^{a}_{sc}\Gamma^{s}_{bd}-\Gamma^{a}_{sd}\Gamma^{s}_{bc}$
are:

$R^{1}{}_{112}=R^{2}{}_{221}=-R^{2}{}_{212}=-R^{1}{}_{121}=-\frac{32b^{2}w}{u^{5}},\enskip
R^{2}{}_{112}=-R^{2}{}_{121}=\frac{24bw}{u^{4}}.$

Solving the generalized KT condition $C_{(ab|c)}=0$, we find that the
connection (117) admits only the second order generalized KT

$C_{ab}=ke^{\frac{12\beta w}{u^{2}}}\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)$ (118)

where $k$ is an arbitrary constant.

Solving the generalized Killing vector (KV) condition $L_{(a|b)}=0$, we find
$L_{a}=0$; therefore, generalized KVs do not exist.

Moreover, it can be shown that non-zero vectors $B_{a}$ which generate
reducible generalized KTs of the form $B_{(a|b)}$ do not exist as well.

Applying Theorem 1 for $m=2$, we find that the system admits only one QFI
which is the

$I=e^{\frac{12\beta w}{u^{2}}}\left(\dot{u}\dot{w}+\frac{1}{12\beta}\right).$
(119)

To prove that the given system is integrable, one needs one more independent
autonomous FI of higher order in involution.

##  9 Application 3: A new superintegrable potential which admits time-
dependent QFIs

In [13], using the separability of the corresponding Hamilton-Jacobi equation
in more than two coordinate systems, all minimally and maximally
superintegrable potentials in the Euclidean space $E^{3}$ that admit
autonomous QFIs are determined. We extend this result to the case where time-
dependent FIs are considered.

Applying Theorem 1 for $m=2$ (QFIs), $q^{a}=(x,y,z)$, $\Gamma^{a}_{bc}=0$ and
$Q^{a}=V^{a,}$, where $V(x,y,z)$ denotes the potential, we find the new
maximally superintegrable potential in $E^{3}$

$V(x,y,z)=-\frac{\lambda^{2}}{2}R^{2}+\frac{kx}{y^{2}R}+\frac{c_{1}}{y^{2}}-\frac{\lambda^{2}}{8}z^{2}+\frac{c_{2}}{z^{2}}$
(120)

where $\lambda\neq 0,k,c_{1},c_{2}$ are arbitrary constants, and
$R=\sqrt{x^{2}+y^{2}}$.

The potential (120) admits the following independent (autonomous and time-
dependent) QFIs:

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)-\frac{\lambda^{2}}{2}R^{2}+\frac{kx}{y^{2}R}+\frac{c_{1}}{y^{2}}-\frac{\lambda^{2}}{8}z^{2}+\frac{c_{2}}{z^{2}}$
(121) $\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}M_{3}^{2}+\frac{(kR+c_{1}x)x}{y^{2}}$ (122)
$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{z}^{2}-\frac{\lambda^{2}}{8}z^{2}+\frac{c_{2}}{z^{2}}$
(123) $\displaystyle I_{4}$ $\displaystyle=$ $\displaystyle e^{\lambda
t}\left[M_{3}(\dot{y}-\lambda
y)+\frac{2c_{1}x}{y^{2}}+\frac{k(y^{2}+2x^{2})}{y^{2}R}\right]$ (124)
$\displaystyle I_{5}$ $\displaystyle=$ $\displaystyle e^{\lambda
t}\left[\left(\dot{z}-\frac{\lambda}{2}z\right)^{2}+\frac{2c_{2}}{z^{2}}\right].$
(125)

The QFI $I_{1}$ is the Hamiltonian of the system and the vector
$M_{i}=\left(y\dot{z}-z\dot{y},z\dot{x}-x\dot{z},x\dot{y}-y\dot{x}\right)$
with $i=1,2,3$ is the angular momentum.

##  10 Application 4: A new superintegrable separable potential which admits
an autonomous CFI

It is well-known that separable Newtonian potentials of the form
$V(x,y)=F_{1}(x)+F_{2}(y)$ admit the QFIs
$J_{1}=\frac{1}{2}\dot{x}^{2}+F_{1}(x)$ and
$J_{2}=\frac{1}{2}\dot{y}^{2}+F_{2}(y)$, where $F_{1}$ and $F_{2}$ are
arbitrary smooth functions of their arguments. The question is if there are
functions $F_{1}$ and $F_{2}$ for which the corresponding potential $V(x,y)$
is superintegrable.

A partial answer to this problem has been given in [14] by considering
autonomous CFIs as the third FI. One of the third order superintegrable
potentials found in [14] is the

$V(x,y)=c_{1}y^{2}+F(x)$ (126)

where $c_{1}$ is an arbitrary non-zero constant and $F(x)$ is an arbitrary
smooth function satisfying the condition

$k_{2}x^{2}+4k_{1}^{2}+\left(9F-c_{1}x^{2}\right)\left(F-c_{1}x^{2}\right)^{3}-4k_{1}\left(F-c_{1}x^{2}\right)\left(3F+c_{1}x^{2}\right)=0$
(127)

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

Using Theorem 1, we generalize the above result and determine a class of
superintegrable potentials which contains the potential (126) as a special
case.

We apply Theorem 1 for $m=3$ (CFIs), $q^{a}=(x,y)$, $\Gamma^{a}_{bc}=0$ and
$Q^{a}=V^{,a}$, where $V(x,y)$ denotes potentials of the form
$V=F_{1}(x)+F_{2}(y)$. We find that the potential (126) is superintegrable due
to the three independent autonomous FIs:

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{x}^{2}+F(x)$ (128) $\displaystyle I_{2}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\dot{y}^{2}+c_{1}y^{2}$ (129)
$\displaystyle I_{3}$ $\displaystyle=$ $\displaystyle L\dot{x}^{2}-\left(3yF-
c_{1}x^{2}y+k_{3}y\right)\dot{x}+\frac{F^{\prime}}{2c_{1}}\left(3F-c_{1}x^{2}+k_{3}\right)\dot{y}$
(130)

where $L\equiv x\dot{y}-y\dot{x}$ is the angular momentum,
$F^{\prime}\equiv\frac{dF}{dx}$ and the function $F(x)$ satisfies the
condition

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
k_{2}x^{2}+4k_{1}^{2}+\left(9F-c_{1}x^{2}\right)\left(F-c_{1}x^{2}\right)^{3}-4k_{1}\left(F-c_{1}x^{2}\right)\left(3F+c_{1}x^{2}\right)+$
(131)
$\displaystyle+4k_{3}\left(3F-c_{1}x^{2}\right)\left(F-c_{1}x^{2}\right)^{2}+4k_{3}^{2}\left(F-c_{1}x^{2}\right)^{2}-\frac{8k_{1}k_{3}}{3}\left(3F-c_{1}x^{2}\right)$

where $k_{1},k_{2},k_{3}$ are arbitrary constants. We note that for $k_{3}=0$
condition (131) reduces to condition (127). Therefore, the superintegrable
potential (C.6) of [14] is a subcase of (126).

##  11 Conclusions

We draw the following conclusions:
a) We have developed a direct systematic method to compute the $m$th-order FIs
of the autonomous (in general non-Riemannian) dynamical systems (1) in terms
of the ‘symmetries’ of the geometric objects (symmetric connection or kinetic
metric, depending on the case) defined by the dynamical equations.
b) This method applies to non-Riemanian geometries with a symmetric
connection. It has been shown that the $m$th-order FIs require the generalized
KTs and KVs defined by the symmetric connection $\Gamma_{bc}^{a}(q)$. The case
of a Riemannian connection is a special case, where the $m$th-order FIs can be
related to a gauged weak generalized Noether symmetry by means of the Inverse
Noether Theorem.
c) The system of PDEs (5) - (8) resulting from the condition
$\frac{dI^{(m)}}{dt}=0$ along the dynamical equations consists of two parts: A
geometric part (eqs. (5) and (6) ) common to all systems which share the same
connection; and a dynamical part (eqs. (7) and (8) ) which includes the
generalized forces $Q^{a}$ of the specific system.
d) We determined the condition which the connection coefficients must satisfy
in order the 2d dynamical systems (86) - (87) to be non-Riemannian.

Obviously, Theorem 1 provides a new systematic way to determine the higher
order FIs, autonomous and time-dependent, of autonomous (in general non-
Riemannian) dynamical systems of the form (1).

## Appendix

We assume that the totally symmetric tensor quantities

$M_{i_{1}...i_{r}}(t,q)=\sum_{N_{r}=0}^{n_{r}}L_{(N_{r})i_{1}...i_{r}}(q)t^{N_{r}},\enskip
r=1,2,...,m,\enskip m\geq 1$ (132)

where $L_{(N_{r})i_{1}...i_{r}}(q)$, $N_{r}=0,1,...,n_{r}$, are arbitrary
$r$-rank totally symmetric tensors and $n_{r}\geq 0$ is the degree of the
polynomial associated with the $r$-rank tensor $M_{i_{1}...i_{r}}(t,q)$. We
note that the degrees $n_{r}$ of the above polynomial expressions of $t$ may
be infinite.

We consider the following cases.

I. Case with $n$ finite.

Substituting (132) in the system of PDEs (5) - (8), we obtain the following
system of polynomial equations in $t$:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{N=0}^{n}L_{(N)(i_{1}...i_{m}|i_{m+1})}t^{N}$ (133)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
M_{,t}-\sum^{n}_{N=0}L_{(N)i_{1}}Q^{i_{1}}t^{N}$ (134) $\displaystyle 0$
$\displaystyle=$ $\displaystyle
M_{,i_{1}}+\sum^{n>0}_{N=1}\left[NL_{(N)i_{1}}-2L_{(N-1)i_{1}i_{2}}Q^{i_{2}}\right]t^{N-1}-2L_{(n)i_{1}i_{2}}Q^{i_{2}}t^{n},\enskip
m>1$ (135) $\displaystyle 0$ $\displaystyle=$ $\displaystyle\sum^{n-1\geq
0}_{N=0}\left[(N+1)L_{(N+1)i_{1}...i_{r}}+L_{(N)(i_{1}...i_{r-1}|i_{r})}-(r+1)L_{(N)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}\right]t^{N}+$
(136)
$\displaystyle+\left[L_{(n)(i_{1}...i_{r-1}|i_{r})}-(r+1)L_{(n)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}\right]t^{n},\enskip
r=2,3,...,m-1,\enskip m>2$ $\displaystyle 0$ $\displaystyle=$ $\displaystyle
M_{,i_{1}}+\sum^{n>0}_{N=1}NL_{(N)i_{1}}t^{N-1},\enskip m=1$ (137)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\sum^{n-1\geq
0}_{N=0}\left[(N+1)L_{(N+1)i_{1}...i_{m}}+L_{(N)(i_{1}...i_{m-1}|i_{m})}\right]t^{N}+L_{(n)(i_{1}...i_{m-1}|i_{m})}t^{n},\enskip
m>1$ (138)

where –without loss of generality– the polynomial expressions (132) of $t$ are
assumed to be of the same degree, that is, $n=n_{r}$ for all values of $r$.
All the results with $n\neq n_{r}$ are derived as subcases from the case
$n=n_{r}$. We note also that: Equation (133) is derived from (5), (134) from
(8), (135) from (7) for $r=1$, (136) from (7) for $r=2,3,...,m-1$, (137) from
(6) for $m=1$, and (138) from (6) for $m>1$.

Equation (133) implies that the quantities $L_{(N)i_{1}...i_{m}}$ with
$N=0,1,...,n$ are $m$th-order generalized KTs. For $m=1$, $L_{(N)i_{1}}$ are
generalized KVs.

Integrating equation (134), we find that

$M=\sum^{n}_{N=0}L_{(N)i_{1}}Q^{i_{1}}\frac{t^{N+1}}{N+1}+G(q)$ (139)

where $G(q)$ is an arbitrary smooth function. _We note that the integrability
conditions of the scalar $M(t,q)$ have been replaced by the integrability
conditions $G_{,[ab]}=0$ of the function $G(q)$._

Replacing $M$ from (139), equation (135) gives:

$\displaystyle 0=$ $\displaystyle
G_{,i_{1}}+L_{(1)i_{1}}(n>0)-2L_{(0)i_{1}i_{2}}Q^{i_{2}}+$
$\displaystyle+\sum^{n-1\geq
1}_{N=1}\frac{1}{N}\left[\left(L_{(N-1)c}Q^{c}\right)_{,i_{1}}+N(N+1)L_{(N+1)i_{1}}-2NL_{(N)i_{1}i_{2}}Q^{i_{2}}\right]t^{N}+$
$\displaystyle+\frac{1}{n}\underbrace{\left[\left(L_{(n-1)c}Q^{c}\right)_{,i_{1}}-2nL_{(n)i_{1}i_{2}}Q^{i_{2}}\right]}_{n>0}t^{n}+\left(L_{(n)c}Q^{c}\right)_{,i_{1}}\frac{t^{n+1}}{n+1},\enskip
m>1\implies$ $\displaystyle G_{,i_{1}}$ $\displaystyle=$ $\displaystyle
2L_{(0)i_{1}i_{2}}Q^{i_{2}}-L_{(1)i_{1}}(n>0),\enskip m>1$ (140)
$\displaystyle\left(L_{(k-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle 2kL_{(k)i_{1}i_{2}}Q^{i_{2}}-k(k+1)L_{(k+1)i_{1}},\enskip
k=1,2,...,n-1,\enskip n>1,\enskip m>1$ (141)
$\displaystyle\left(L_{(n-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle 2nL_{(n)i_{1}i_{2}}Q^{i_{2}},\enskip n>0,\enskip m>1$ (142)
$\displaystyle L_{(n)i_{1}}Q^{i_{1}}$ $\displaystyle=$ $\displaystyle
s_{0},\enskip m>1$ (143)

where $s_{0}$ is an arbitrary constant. _The notation $L_{(1)i_{1}}(n>0)$
indicates that the vector $L_{(1)i_{1}}$ exists only when the degree of the
polynomial $n>0$, that is, when $n=0$, the vector $L_{(1)i_{1}}$ vanishes_.

Replacing (139) in (137), we find the following conditions:

$\displaystyle G_{,i_{1}}$ $\displaystyle=$ $\displaystyle-
L_{(1)i_{1}}(n>0),\enskip m=1$ (144)
$\displaystyle\left(L_{(k-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle-k(k+1)L_{(k+1)i_{1}},\enskip k=1,2,...,n-1,\enskip n>1,\enskip
m=1$ (145) $\displaystyle L_{(n-1)i_{1}}Q^{i_{1}}$ $\displaystyle=$
$\displaystyle s_{1},\enskip n>0,\enskip m=1$ (146) $\displaystyle
L_{(n)i_{1}}Q^{i_{1}}$ $\displaystyle=$ $\displaystyle s_{0},\enskip m=1$
(147)

where $s_{1}$ is an arbitrary constant.

We observe that conditions (144) - (147) are emerged from conditions (140) -
(143) if we rewrite the latter in the following compact form:

$\displaystyle G_{,i_{1}}$ $\displaystyle=$ $\displaystyle
2L_{(0)i_{1}i_{2}}(m>1)Q^{i_{2}}-L_{(1)i_{1}}(n>0)$ (148)
$\displaystyle\left(L_{(k-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle 2kL_{(k)i_{1}i_{2}}(m>1)Q^{i_{2}}-k(k+1)L_{(k+1)i_{1}},\enskip
k=1,2,...,n-1,\enskip n>1$ (149)
$\displaystyle\left(L_{(n-1)c}Q^{c}\right)_{,i_{1}}$ $\displaystyle=$
$\displaystyle 2nL_{(n)i_{1}i_{2}}(m>1)Q^{i_{2}},\enskip n>0$ (150)
$\displaystyle L_{(n)i_{1}}Q^{i_{1}}$ $\displaystyle=$ $\displaystyle s_{0}.$
(151)

_The notation $L_{(0)i_{1}i_{2}}(m>1)$ indicates that the quantities
$L_{(0)i_{1}i_{2}}$ exist only when the degree of the FI $m>1$, that is, when
$m=1$, the quantities $L_{(0)i_{1}i_{2}}$ vanish._

Conditions (149) and (150) are written compactly as follows:

$\left(L_{(k-1)c}Q^{c}\right)_{,i_{1}}=2kL_{(k)i_{1}i_{2}}(m>1)Q^{i_{2}}-k(k+1)L_{(k+1)i_{1}}(k<n),\enskip
k=1,2,...,n,\enskip n>0.$ (152)

_The notation $L_{(k+1)i_{1}}(k<n)$ indicates that the vector $L_{(k+1)i_{1}}$
exists only when $k<n$, that is, if $k\geq n$, the vector $L_{(k+1)i_{1}}$
vanishes._

Equation (136) implies that:

$\displaystyle L_{(k)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(k)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(k+1)L_{(k+1)i_{1}...i_{r}},$
(153) $\displaystyle k=0,1,...,n-1,\enskip r=2,3,...,m-1,\enskip n>0,\enskip
m>2$ $\displaystyle L_{(n)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(n)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}},\enskip
r=2,3,...,m-1,\enskip m>2.$ (154)

Conditions (153) and (154) are written compactly as follows:

$\displaystyle L_{(k)(i_{1}...i_{r-1}|i_{r})}$ $\displaystyle=$
$\displaystyle(r+1)L_{(k)i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-(k+1)L_{(k+1)i_{1}...i_{r}}(k<n),$
(155) $\displaystyle k=0,1,...,n,\enskip r=2,3,...,m-1,\enskip m>2.$

_The notation $L_{(k+1)i_{1}...i_{r}}(k<n)$ indicates that the quantities
$L_{(k+1)i_{1}...i_{r}}$ exist only when $k<n$, that is, if $k\geq n$, the
quantities $L_{(k+1)i_{1}...i_{r}}$ vanish._

Equation (138) implies that the quantities $L_{(n)i_{1}...i_{m-1}}$ with $m>1$
are the components of an $(m-1)$th-order generalized KT and the $m$th-order
generalized KTs

$L_{(k)i_{1}...i_{m}}=-\frac{1}{k}L_{(k-1)(i_{1}...i_{m-1}|i_{m})},\enskip
k=1,2,...,n,\enskip n>0,\enskip m>1.$ (156)

The $m$th-order FI (2) is

$I^{(m)}_{n}=\sum_{r=1}^{m}\left(\sum^{n}_{N=0}L_{(N)i_{1}...i_{r}}t^{N}\right)\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+s_{0}\frac{t^{n+1}}{n+1}+\sum^{n>0}_{N=1}L_{(N-1)c}Q^{c}\frac{t^{N}}{N}+G(q)$
(157)

where $m\geq 1$, $n\geq 0$, $L_{(N)i_{1}...i_{m}}(q)$ with $N=0,1,...,n$ are
$m$th-order generalized KTs satisfying the condition (156),
$L_{(n)i_{1}...i_{m-1}}(q)$ with $m>1$ is an $(m-1)$th-order generalized KT
and the constant $s_{0}$ is given by (151). The function $G(q)$ and the
totally symmetric tensors $L_{(N)i_{1}...i_{r}}(q)$ satisfy the conditions
(148), (152) and (155).

The notation $I^{(m)}_{n}$ means the $m$th-order FI (upper index) with time-
dependence $n$ (lower index). For example, the $I^{(2)}_{n}$ is a QFI whose
coefficients are expressed as polynomials of $t$ of degree fixed by $n$.

II. Case with $n$ infinite.

The polynomial expressions (132) as $n_{r}=n\to\infty$ turn into the infinite
sum (series)

$M_{i_{1}...i_{r}}(t,q)=\sum_{N=0}^{\infty}L_{(N)i_{1}...i_{r}}(q)t^{N},\enskip
r=1,2,...,m,\enskip m\geq 1.$ (158)

It is found that new results –different from those found with $n$ finite– are
derived in the case that

$L_{(N)i_{1}...i_{r}}(q)=\frac{\lambda_{r}^{N}}{N!}L_{i_{1}...i_{r}}(q)$ (159)

where $\lambda_{r}$ are arbitrary non-zero constants and
$L_{i_{1}...i_{r}}(q)$ are $r$-rank totally symmetric tensors.

Replacing (159) in (158), we find

$M_{i_{1}...i_{r}}(t,q)=L_{i_{1}...i_{r}}(q)\sum_{N=0}^{\infty}\frac{(\lambda_{r}t)^{N}}{N!}=e^{\lambda_{r}t}L_{i_{1}...i_{r}}(q).$
(160)

Substituting (160) in the system of PDEs (5) - (8), we obtain the following
system of equations:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle L_{(i_{1}...i_{m}|i_{m+1})}$
(161) $\displaystyle 0$ $\displaystyle=$ $\displaystyle M_{,t}-e^{\lambda
t}L_{c}Q^{c}$ (162) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
M_{,i_{1}}+e^{\lambda t}\left(\lambda
L_{i_{1}}-2L_{i_{1}i_{2}}Q^{i_{2}}\right),\enskip m>1$ (163) $\displaystyle 0$
$\displaystyle=$ $\displaystyle L_{(i_{1}...i_{r-1}|i_{r})}+\lambda
L_{i_{1}...i_{r}}-(r+1)L_{i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}},\enskip
r=2,3,...,m-1,\enskip m>2$ (164) $\displaystyle 0$ $\displaystyle=$
$\displaystyle M_{,i_{1}}+\lambda e^{\lambda t}L_{i_{1}},\enskip m=1$ (165)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\lambda
L_{i_{1}...i_{m}}+L_{(i_{1}...i_{m-1}|i_{m})},\enskip m>1$ (166)

where –without loss of generality– all the non-zero constants $\lambda_{r}$
are fixed to the same non-zero constant $\lambda$. The $m$th-order FI produced
from this assumption contains as subcases all the FIs associated with
constants $\lambda_{r}$ which are not all the same.

Equation (161) implies that $L_{i_{1}...i_{m}}$ is an $m$th-order generalized
KT.

Integrating equation (162), we find

$M=\frac{e^{\lambda t}}{\lambda}L_{c}Q^{c}+G(q)$ (167)

where $G(q)$ is an arbitrary smooth function.

Replacing (167) in equations (163) and (165), we find that $G(q)=const\equiv
0$ and the condition:

$\left(L_{c}Q^{c}\right)_{,i_{1}}=2\lambda
L_{i_{1}i_{2}}(m>1)Q^{i_{2}}-\lambda^{2}L_{i_{1}}.$ (168)

Equation (164) gives the condition

$L_{(i_{1}...i_{r-1}|i_{r})}=(r+1)L_{i_{1}...i_{r}i_{r+1}}Q^{i_{r+1}}-\lambda
L_{i_{1}...i_{r}},\enskip r=2,3,...,m-1,\enskip m>2$ (169)

while the remaining condition (166) implies that

$L_{i_{1}...i_{m}}=-\frac{1}{\lambda}L_{(i_{1}...i_{m-1}|i_{m})},\enskip m>1.$
(170)

The associated $m$th-order FI (2) is

$I^{(m)}_{e}=\frac{e^{\lambda
t}}{\lambda}\left(\lambda\sum^{m}_{r=1}L_{i_{1}...i_{r}}\dot{q}^{i_{1}}...\dot{q}^{i_{r}}+L_{c}Q^{c}\right)$
(171)

where $\lambda\neq 0$, $L_{i_{1}...i_{m}}(q)$ is an $m$th-order generalized KT
satisfying the condition (170), and the remaining totally symmetric tensors
$L_{i_{1}...i_{r}}(q)$ with $r=1,2,...,m-1$ and $m>1$ satisfy the conditions
(168) and (169).

The notation $I^{(m)}_{e}$ indicates the $m$th-order FI (upper index) with
exponential time-dependence (lower index).

The above complete the proof of Theorem 1.

## Acknowledgments

We thank Dr Andronikos Paliathanasis for proposing the dynamical system (115)
- (116).

## Data Availability

The data that supports the findings of this study are available within the
article.

## Conflict of interest

The authors declare no conflict of interest.

## Author contributions

Conceptualization, Antonios Mitsopoulos and Michael Tsamparlis; Formal
analysis, Antonios Mitsopoulos; Methodology, Antonios Mitsopoulos and Michael
Tsamparlis; Writing – original draft, Antonios Mitsopoulos; Writing – review
and editing, Antonios Mitsopoulos, Michael Tsamparlis and Aniekan Magnus
Ukpong.

## References

  * [1] V.I. Arnold, _‘Mathematical Methods of Classical Mechanics’_ , Springer, (1989), proof in pp. 272-284.
  * [2] E.T. Whittaker, _‘A Treatise on the Analytical Dynamics of Particles and Rigid Bodies’_ , 2nd ed., Cambridge University Press, (1917).
  * [3] G.H. Katzin and J. Levine, _‘Geodesic first integrals with explicit path-parameter dependence in Riemannian space-times’_ , J. Math. Phys. 22(9), 1878 (1981).
  * [4] G. Thompson, _‘Polynomial constants of motion in flat space’_ , J. Math. Phys. 25(12), 3474 (1984).
  * [5] J.T. Horwood, _‘Higher order first integrals in classical mechanics’_ , J. Math. Phys 48, 102902 (2007).
  * [6] S. Post and P. Winternitz, _‘General Nth order integrals of motion in the Euclidean plane’_ , J. Phys. A: Math. Gen. 48, 405201 (2015).
  * [7] A. Mitsopoulos and M. Tsamparlis, _‘Higher order first integrals of autonomous dynamical systems’_ , J. Geom. Phys. 170, 104383 (2021).
  * [8] D.S. Djukic and B.D. Vujanovic, _‘Noether’s Theory in Classical Nonconservative Mechanics’_ , Acta Mechanica 23, 17 (1975).
  * [9] M. Tsamparlis and A. Mitsopoulos, _‘First integrals of holonomic systems without Noether symmetries’_ , J. Math. Phys. 61, 122701 (2020).
  * [10] A. Mitsopoulos and M. Tsamparlis, _‘Quadratic first integrals of time-dependent dynamical systems of the form $\ddot{q}^{a}=-\Gamma^{a}_{bc}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$’_, Mathematics 9(13), 1503 (2021).
  * [11] N. Dimakis, P.A. Terzis and T. Christodoulakis, _‘Integrability of geodesic motions in curved manifolds through nonlocal conserved charges’_ , Phys. Rev. D 99, 104061 (2019).
  * [12] A. Mitsopoulos and M. Tsamparlis, _‘Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy’_ , Symmetry 14(9), 1870 (2022).
  * [13] N.W. Evans, _‘Superintegrability in classical mechanics’_ , Phys. Rev. A 41(10), 5666 (1990).
  * [14] S. Gravel, _‘Hamiltonians separable in Cartesian coordinates and third-order integrals of motion’_ , J. Math. Phys. 45(3), 1003 (2004).