# Integrable and Superintegrable Extensions
of the Rational Calogero-Moser Model in 3 Dimensions

Allan P. Fordy and Qing Huang School of Mathematics, University of Leeds,
Leeds LS2 9JT, UK. E-mail<EMAIL_ADDRESS>of Mathematics, Center
for Nonlinear Studies, Northwest University, Xi’an 710069, People’s Republic
of China E-mail<EMAIL_ADDRESS>
###### Abstract

We consider a class of Hamiltonian systems in 3 degrees of freedom, with a
particular type of quadratic integral and which includes the rational
Calogero-Moser system as a particular case. For the general class, we
introduce separation coordinates to find the general separable (and therefore
Liouville integrable) system, with two quadratic integrals. This gives a
coupling of the Calogero-Moser system with a large class of potentials,
generalising the series of potentials which are separable in parabolic
coordinates. Particular cases are superintegrable, including Kepler and a
resonant oscillator.

The initial calculations of the paper are concerned with the flat (Cartesian
type) kinetic energy, but in Section 5, we introduce a conformal factor
$\varphi$ to $H$ and extend the two quadratic integrals to this case. All the
previous results are generalised to this case. We then introduce some 2 and 3
dimensional symmetry algebras of the Kinetic energy (Killing vectors), which
restrict the conformal factor. This enables us to reduce our systems from 3 to
2 degrees of freedom, giving rise to many interesting systems, including both
Kepler type and Hénon-Heiles type potentials on a Darboux-Koenigs $D_{2}$
background.

Keywords: Hamiltonian system, super-integrability, Poisson algebra, conformal
algebra, Calogero-Moser system, Kepler problem, Darboux-Koenigs metric, Hénon-
Heiles system.

MSC: 17B63, 37J15, 37J35,70G45, 70G65, 70H06

##  1 Introduction

The Calogero-Moser system is the archetypal integrable, many-body problem,
both classical and quantum [2, 19], and is known to be maximally
superintegrable [22, 18]. These authors derive the first integrals from the
Lax matrix, but here we wish to use methods which can be used, regardless of
whether or not a Lax matrix is known. In this paper we only consider the case
of 3 degrees of freedom.

The Calogero-Moser system in 3 degrees of freedom is

$H=\frac{1}{2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})+g^{2}\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right),$
(1a) which can be obtained from a Lax matrix $L$, as
$\frac{1}{2}\mbox{tr}L^{2}$. In the Lax approach, a full set of Poisson
commuting integrals (for general $n$ degrees of freedom) can be written as
$\frac{1}{k}\mbox{tr}L^{k}$. Extending this approach, Wojciechowski [22] found
more integrals, some of which are quadratic and related to the ones given
below. Whilst these Poisson commute with $H$, they generate a non-commutative
Poisson algebra, as is necessarily the case for superintegrable systems.

In this paper we particularly focus on one particular quadratic integral,
given in the construction of [22], which can be written

$X_{1}=(p_{1}+p_{2}+p_{3})S-2(q_{1}+q_{2}+q_{3})H,$ (1b)

where $S$ is a conformal element, given by (58b) in the Appendix (Section
10.1). In formula (1b), $H$ just denotes (1a), but in Section 2 we consider
$H$ with a general potential $U$, and ask for a modification of $X_{1}$
(deformed with its own “potential” term) to be a first integral.

Indeed, in Section 2 we classify the functions $U$, which allow such an
integral $X_{1}$, by reducing the system to separation variables, $(x,y,z)$,
which are a 3-dimensional extension of the standard parabolic coordinates of
the plane (labelled here $(x,y)$). Hence these potentials (in the $(x,y)$
components) are just the standard sequence (see Chapter 2 of [20]), including
the Kepler problem, the KdV case of integrable Hénon-Heiles potential, and
many more. The $z$ component allows for the connection to the Calogero-Moser
system (when re-written in the original $q_{i}$ coordinates). Separation of
variables gives a third quadratic function $X_{2}$ and the three functions
$H,\,X_{1},\,X_{2}$ play an important role throughout the paper.

To construct $(x,y,z)$, we first introduce an intermediate Cartesian
coordinate system $Q_{i}$, in which many of these potentials take their most
natural form.

Whilst Section 2 discusses the general structure of Liouville integrability in
the three coordinate systems, $(x,y,z),\;(Q_{1},Q_{2},Q_{3})$ and
$(q_{1},q_{2},q_{3})$, Section 3 is concerned with special potentials in the
$(Q_{1},Q_{2},Q_{3})$ coordinates. Whilst the general system is simply
Liouville integrable, many of these special cases have additional integrals,
making them superintegrable. It’s in these coordinates that such potentials as
Kepler, a resonant oscillator and Hénon-Heiles arise.

In Section 4 the Calogero-Moser system is specifically considered with both
Kepler and oscillator coupling.

Up to this point the kinetic energy has been of the standard flat, Cartesian
type, but in Section 5, we introduce a conformal factor $\varphi$ to the
entire separable Hamiltonian $H$ (in $(x,y,z)$ coordinates), requiring it to
continue to be separable in these coordinates. This modifies the definition of
$X_{1}$ and $X_{2}$, but these three functions remain in involution. The
remaining sections are concerned with this extension, mainly in terms of the
coordinates $Q_{i}$.

Section 6 extends the cases of Section 3 to conformally flat metrics. Whilst
the restriction of the potential is allowed without any further constraints on
the conformal factor $\varphi$, demanding additional first integrals, as in
the superintegrable cases, does force constraints on $\varphi$. In this
section we see extensions of Kepler, resonant oscillator and Hénon-Heiles
potentials. In particular, one of the extensions to the previous resonant
oscillator is a superintegrable, 3D extension of the Darboux-Koenigs $D_{1}$
kinetic energy with potential (see Section 6.3.2).

In Section 7 we introduce some 2 and 3 dimensional symmetry algebras (linear
functions of momenta, which commute with the kinetic energy of $H$), discussed
in detail in [12, 13]. The existence of such symmetries restricts the
conformal factor $\varphi$. We can then restict the entire Hamiltonian with
just one of these symmetries, after which we can choose coordinates in which
the Hamiltonian can be interpreted as a 2 dimensional reduction.

Many interesting reduced systems arise in this way. For example the
Hamiltonian (33a) is a Kepler problem on a Darboux-Koenigs $D_{2}$ background.
It is superintegrable, with two independent quadratic integrals. Similarly,
the Hamiltonian (34a) is of generalised Hénon-Heiles type on a Darboux-Koenigs
$D_{2}$ background. It has one quadratic integral, so is Liouville integrable
(but not superintegrable). The Hamiltonian (48c) is interesting in the context
of Darboux-Koenigs theory; it has a Darboux-Koenigs $D_{3}$ kinetic energy,
but the given potential only allows one quadratic integral. However, there is
an independent quartic integral, so the system is maximally superintegrable
and outside the class considered in [16].

In Section 8, we present the Kepler-Calogero-Moser system with a particular
conformally flat metric.

Commonly used notations are defined in the Appendix (Section 10.1).

##  2 The General Potential Compatible with $X_{1}$: Flat Metric

We now consider the two functions

$H=\frac{1}{2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})+U(q_{1},q_{2},q_{3}),\quad
X_{1}=(p_{1}+p_{2}+p_{3})S-2(q_{1}+q_{2}+q_{3})H+V_{1}(q_{1},q_{2},q_{3}),$
(2)

with $X_{1}$ an extension of (1b). We have already seen that these functions
commute for the special choices (1a) and (1b). We now seek conditions on the
general $U$ and $V_{1}$ of (2), such that $\\{H,X_{1}\\}=0$.

Since the leading parts of these functions are quadratic in momenta, we can
use separation of variables, which corresponds to simultaneous diagonalisation
of the defining quadratic forms by a canonical transformation. We do this in
two steps. The intermediate coordinate system is not only convenient, but of
interest in itself.

###  2.1 Intermediate Transformation to $(Q_{i},P_{i})$

We first perform an orthogonal transformation, with $Q_{1}$ the centre of mass
coordinate:

$Q_{1}=\frac{1}{\sqrt{3}}(q_{1}+q_{2}+q_{3}),\quad
Q_{2}=\frac{1}{\sqrt{2}}(q_{1}-q_{2}),\quad
Q_{3}=\frac{1}{\sqrt{6}}(q_{1}+q_{2}-2q_{3}),$ (3)

in which

$p_{1}+p_{2}+p_{3}=\sqrt{3}P_{1},\quad
p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=P_{1}^{2}+P_{2}^{2}+P_{3}^{2},\quad
S=Q_{1}P_{1}+Q_{2}P_{2}+Q_{3}P_{3}.$

Therefore (up to an overall factor of $\sqrt{3}$ dropped from $X_{1}$), we
obtain

$H=\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})+\bar{U}(Q_{1},Q_{2},Q_{3}),\quad
X_{1}=P_{1}S-2Q_{1}H+\bar{V}_{1}(Q_{1},Q_{2},Q_{3}),$ (4a) where we see that
$X_{1}$ can be written in the same form as (2).

When written explicitly, the quadratic part of $X_{1}$ is now simpler:

$X_{1}^{0}=P_{1}(Q_{2}P_{2}+Q_{3}P_{3})-Q_{1}(P_{2}^{2}+P_{3}^{2}).$ (4b)

The eigenvectors of the corresponding quadratic form are each defined up to an
overall functional coefficient, which can be used to write it as the gradient
of a new coordinate function.

###  2.2 Separation of Variables

The details of the calculation are unimportant, but lead to the change of
coordinates:

$x=\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}+Q_{1},\quad
y=\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}-Q_{1},\quad z=\frac{Q_{3}}{Q_{2}},$ (5)

which reduce to the standard two dimensional parabolic coordinates when
$Q_{3}=0$. In these coordinates, the kinetic energy
$H^{0}=\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})$ takes the form

$H^{0}=\frac{2xp_{x}^{2}+2yp_{y}^{2}}{x+y}+\frac{(1+z^{2})^{2}p_{z}^{2}}{2xy},$
(6a) with the pure $x-y$ part being the standard 2D parabolic coordinates
case.

We can then add the standard separable potentials to (6a):

$H=\frac{2xp_{x}^{2}+2yp_{y}^{2}+A_{1}(x)+A_{2}(y)}{x+y}+\frac{(1+z^{2})^{2}p_{z}^{2}+2(1+z^{2})B(z)}{2xy},$
(6b)

without destroying commutativity.

###### Remark 2.1

We could have defined $z$ as $\arctan\left(\frac{Q_{3}}{Q_{2}}\right)$ to make
the $z$ part just $\frac{p_{z}^{2}}{2xy}$, but our main interest is in the
$Q_{i}$ and $q_{i}$ coordinates, so such refinements are unnecessary. The
definition of $B(z)$ incorporates the factor $(1+z^{2})$ to simplify the
expression in the $Q_{i}$ coordinates.

Separation of variables then leads to 3 quadratic commuting integrals:

$H=E\quad\Rightarrow\quad
2xp_{x}^{2}+A_{1}(x)+\frac{\gamma}{2x}-Ex+2yp_{y}^{2}+A_{2}(y)+\frac{\gamma}{2y}-Ey=0,$
(6c)

where $\gamma$ is defined by

$X_{2}=(1+z^{2})^{2}p_{z}^{2}+2(1+z^{2})B(z)=\gamma.$ (6d)

Defining $\alpha=2xp_{x}^{2}+A_{1}(x)+\frac{\gamma}{2x}-Ex$, we can eliminate
$E$ to obtain

$\alpha=\frac{2xy(p_{x}^{2}-p_{y}^{2})+yA_{1}(x)-xA_{2}(y)}{x+y}+\frac{\gamma(y-x)}{2xy},$
(6e)

with $\gamma$ given by (6d). Applying the transformation (5) to $X_{1}$ of
(4a), we see that $\alpha=X_{1}$.

######  Remark 2.2 (Jacobi’s Theorem)

As guaranteed by Jacobi’s Theorem, separability has led to a third quadratic
first integral $X_{2}$, and that $H,\,X_{1}$ and $X_{2}$ are in involution.

###  2.3 Returning to $(Q_{i},P_{i})$ Coordinates

Returning to these coordinates, we have

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})+\frac{A_{1}\left(x\right)+A_{2}\left(y\right)}{2R}+\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(7a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-2Q_{1}H+\frac{xA_{1}(x)-yA_{2}(y)}{2R},$ (7b) $\displaystyle X_{2}$
$\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2(Q_{2}^{2}+Q_{3}^{2})\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(7c) where $x=R+Q_{1},\;y=R-Q_{1}$, with
$R=\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}$.

###  2.4 Returning to $(q_{i},p_{i})$ Coordinates

We can finally return to the original coordinates:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})+\frac{A_{1}(x)+A_{2}(y)}{2\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}}+\frac{W\left(\frac{q_{2}-q_{3}}{q_{1}-q_{2}}\right)}{(q_{1}-q_{2})^{2}},$
(8a) $\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle(p_{1}+p_{2}+p_{3})S-2(q_{1}+q_{2}+q_{3})H+\frac{xA_{1}(x)-yA_{2}(y)}{2R},$
(8b) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(J_{1}+J_{2}+J_{3})^{2}+4(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}-q_{1}q_{2}-q_{2}q_{3}-q_{3}q_{1})\,\frac{W\left(\frac{q_{2}-q_{3}}{q_{1}-q_{2}}\right)}{(q_{1}-q_{2})^{2}},$
(8c) where
$x=\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}+\frac{q_{1}+q_{2}+q_{3}}{\sqrt{3}},\;\;y=\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}-\frac{q_{1}+q_{2}+q_{3}}{\sqrt{3}},\;\;W\left(\frac{q_{2}-q_{3}}{q_{1}-q_{2}}\right)=2B\left(\frac{q_{1}+q_{2}-2q_{3}}{\sqrt{3}(q_{1}-q_{2})}\right),$
and $J_{i}$ are given in (58a).

##  3 Some Specific Potentials in $(Q_{i},P_{i})$ Coordinates

Since the coordinates of (6b) are an extension of the 2 dimensional parabolic
coordinates, the $(x,y)$ part of the potential contains several well known,
important potentials, such as Kepler and one of the Hénon-Heiles cases, which
are now extended to 3 dimensions. To appreciate this it is best to represent
them in the $(Q_{i},P_{i})$ coordinates. The $z$ part of the potential (6b)
adds further extensions to these known potentials in the $(Q_{i},P_{i})$
coordinates, but is most interesting in the $(q_{i},p_{i})$ coordinates, since
here it includes the 3 component Calogero-Moser potential as a special case.
All of these cases are completely integrable, but some have additional
integrals, making them superintegrable.

###  3.1 Special Choices of the Functions $A_{i}(Q_{1},Q_{2},Q_{3})$

Clearly, by setting $A_{1}(x)=A_{2}(y)=-\mu$, the potential of (7a) includes a
Kepler part

$H=\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})-\,\frac{\mu}{\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}}+\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}}.$
(9a) The “$A$” part of the potential in (7a) also includes an infinite family
of polynomial potentials, which directly generalise the well known 2
dimensional case (see, for example, [20]):
$U_{n}=k\,\frac{\left(\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}+Q_{1}\right)^{n+1}+(-1)^{n}\left(\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}-Q_{1}\right)^{n+1}}{2\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}},$
(9b) with $\displaystyle U_{1}=2kQ_{1},\quad
U_{2}=k(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}),\quad
U_{3}=4kQ_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}),$ $\displaystyle
U_{4}=k(16Q_{1}^{4}+12Q_{1}^{2}(Q_{2}^{2}+Q_{3}^{2})+(Q_{2}^{2}+Q_{3}^{2})^{2}).$
(9c)

Hamiltonian (7a), with $U_{2}$ gives case 6 of Table II of [6]. Hamiltonian
(7a), with a combination of $U_{2}$ and $U_{3}$ gives a 3 dimensional version
of the Hénon-Heiles system, with the addition of the
$\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}}$ term, which generalises
the usual $\frac{c}{Q_{2}^{2}}$ term, which appears in the 2 dimensional case.
Hamiltonian (7a), with $U_{4}$ gives a generalisation of case 4 of Table 1 of
[4].

###  3.2 The Case $A_{i}=0$

This is a well known case [1, 21, 15, 3] with 4 independent integrals for
general $B\left(\frac{Q_{3}}{Q_{2}}\right)$. Special cases of this (including
the Calogero-Moser case) allow an algebra with rank 5 (which is maximal in 3
degrees of freedom).

As well as $H,X_{1},X_{2}$, we also have $P_{1}$ and $\Omega$ (an extension of
$J^{2}$, given in the Appendix):

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})+\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(10a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle P_{1}S-2Q_{1}H,$
(10b) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2(Q_{2}^{2}+Q_{3}^{2})\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(10c) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle(Q_{1}P_{2}-Q_{2}P_{1})^{2}+(Q_{2}P_{3}-Q_{3}P_{2})^{2}+(Q_{3}P_{1}-Q_{1}P_{3})^{2}+2(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}}.$
(10d) $\displaystyle=$ $\displaystyle
2(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})H-(Q_{1}P_{1}+Q_{2}P_{2}+Q_{3}P_{3})^{2}.$ The
non-zero Poisson brackets are
$\\{P_{1},X_{1}\\}=2H-P_{1}^{2},\;\;\\{P_{1},\Omega\\}=2X_{1},\;\;\\{X_{1},\Omega\\}=-2P_{1}\Omega,$
(10e) and these 5 functions satisfy the constraint $X_{1}^{2}=2(\Omega-
X_{2})H-P_{1}^{2}\Omega.$ (10f)

###  3.3 The Kepler Case

As well as $H,X_{1},X_{2}$, we also have $\Omega$ (an extension of (58c)):

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})-\,\frac{\mu}{\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}}+\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(11a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-2Q_{1}H-\frac{\mu Q_{1}}{\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}},$ (11b)
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2(Q_{2}^{2}+Q_{3}^{2})\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(11c) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle(Q_{1}P_{2}-Q_{2}P_{1})^{2}+(Q_{2}P_{3}-Q_{3}P_{2})^{2}+(Q_{3}P_{1}-Q_{1}P_{3})^{2}+2(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}}.$
(11d) $\displaystyle=$ $\displaystyle
2(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})H-(Q_{1}P_{1}+Q_{2}P_{2}+Q_{3}P_{3})^{2}+2\mu\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}.$
As well as $\\{X_{1},X_{2}\\}=\\{X_{2},\Omega\\}=0$, we have
$\\{X_{1},\Omega\\}=-2X_{3}$, where
$X_{3}=P_{1}\Omega+\frac{\mu}{\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}}\,\left(Q_{2}(Q_{1}P_{2}-Q_{2}P_{1})+Q_{3}(Q_{1}P_{3}-Q_{3}P_{1})\right),$
(11e) with $\\{\Omega,X_{3}\\}=-2X_{1}\Omega$. We also have
$\\{X_{1},X_{3}\\}=2(X_{2}-2\Omega)H+X_{1}^{2}-\mu^{2},\quad
X_{3}^{2}=\Omega(2\Omega H-2X_{2}H-X_{1}^{2})+\mu^{2}(\Omega-X_{2}).$

###  3.4 The Oscillator Potential $U_{2}$

We start with the 3 integrals related to the separation of variables:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})+\frac{1}{2}\omega^{2}\,(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(12a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-2Q_{1}H+2\omega^{2}\,Q_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}),$ (12b)
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2(Q_{2}^{2}+Q_{3}^{2})\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}}.$
(12c) Since $B$ is independent of $Q_{1}$, we see that we have an additional
integral: $F_{1}=P_{1}^{2}+4\omega^{2}Q_{1}^{2}.$ (12d) We can then define
$F_{2}$ by $\\{X_{1},F_{1}\\}=-2F_{2}\quad\Rightarrow\quad
F_{2}=P_{1}(2H-P_{1}^{2})+2\omega^{2}(2Q_{1}(Q_{2}P_{2}+Q_{3}P_{3})-P_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})).$
(12e) The remaining nonzero Poisson brackets are
$\\{X_{1},F_{2}\\}=(F_{1}-2H)(2H-3F_{1})+4\omega^{2}X_{2},\;\;\\{F_{1},F_{2}\\}=-8\omega^{2}X_{1}.$
(12f) These integrals have rank $4$, satisfying
$F_{2}^{2}=F_{1}(F_{1}-2H)^{2}-4\omega^{2}(F_{1}X_{2}+X_{1}^{2}).$ (12g)

####  3.4.1 Relation to Resonant Oscillator

When $B=0$, this is just a 3 dimensional resonant oscillator, which can be
thought of as a combination of 2 component oscillators, with resonances
$(2,1),\,(2,1)$ and $(1,1)$.

The 2 dimensional oscillator with resonances $(2,1)$ has quadratic and cubic
integrals. We can write $X_{1}$ and $F_{2}$ as the sum of integrals in the
$(Q_{1},Q_{2})$ and $(Q_{1},Q_{3})$ spaces, together with an extra piece
involving $B$:

$\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle(-Q_{1}P_{2}^{2}+Q_{2}P_{1}P_{2}+\omega^{2}Q_{1}Q_{2}^{2})+(-Q_{1}P_{3}^{2}+Q_{3}P_{1}P_{3}+\omega^{2}Q_{1}Q_{3}^{2})-2Q_{1}\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(13a) $\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle(P_{1}P_{2}^{2}-\omega^{2}Q_{2}^{2}P_{1}+4\omega^{2}Q_{1}Q_{2}P_{2})+(P_{1}P_{3}^{2}-\omega^{2}Q_{3}^{2}P_{1}+4\omega^{2}Q_{1}Q_{3}P_{3})+2P_{1}\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(13b) The integral $X_{2}$ is itself related to the $(1,1)$ resonance, as is
another quadratic integral:
$F_{3}^{(2)}=P_{2}^{2}+\omega^{2}Q_{2}^{2}+V_{3}^{(2)}(Q_{2}),$ (13c) where
the additional term $V_{3}^{(2)}(Q_{2})$ is to be determined. The condition
$\\{H,F_{3}^{(2)}\\}=0$ leads to
$B=Q_{2}^{2}\left(\frac{b_{1}}{Q_{2}^{2}}+\frac{b_{2}}{Q_{3}^{2}}\right),\quad
V_{3}^{(2)}=\frac{2b_{1}}{Q_{2}^{2}},$ (13d) giving (12a) as the last case in
Table I of [6]. We see that there is an additional integral
$F_{3}^{(3)}=P_{3}^{2}+\omega^{2}Q_{3}^{2}+\frac{2b_{2}}{Q_{3}^{2}}.$ (13e) We
also have:
$X_{1}=P_{1}S-2Q_{1}H+2\omega^{2}Q_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}),\quad
X_{2}=(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2(Q_{2}^{2}+Q_{3}^{2})\left(\frac{b_{1}}{Q_{2}^{2}}+\frac{b_{2}}{Q_{3}^{2}}\right),$
(13f)

with $F_{1},\,F_{3}^{(2)},\,F_{3}^{(3)},\,X_{1},\,X_{2}$ being rank 5. They
generate a 10 dimensional Poisson algebra, with 5 polynomial constraints, all
of which can be derived as a restriction of the conformally flat case,
discussed in Sections 6.3.2 and 10.2. This algebra is discussed in Section
10.3 of the Appendix.

###  3.5 The Generalised Hénon-Heiles Potential $U_{3}$

We have the three functions in involution:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})+\frac{1}{2}\omega^{2}\,(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+4kQ_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(14a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-2Q_{1}H+2\omega^{2}\,Q_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})$ (14b)
$\displaystyle\hskip
170.71652pt+k(16Q_{1}^{4}+12Q_{1}^{2}(Q_{2}^{2}+Q_{3}^{2})+(Q_{2}^{2}+Q_{3}^{2})^{2})$
$\displaystyle=$ $\displaystyle
P_{3}J_{2}-P_{2}J_{3}+\omega^{2}Q_{1}(Q_{2}^{2}+Q_{3}^{2})+k(Q_{2}^{2}+Q_{3}^{2})^{2}(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})-2Q_{1}\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2(Q_{2}^{2}+Q_{3}^{2})\,\frac{B\left(\frac{Q_{3}}{Q_{2}}\right)}{Q_{2}^{2}},$
(14c) so it is completely integrable. This is a 3 degrees of freedom
generalisation of the standard “KdV case” of integrable Hénon-Heiles models
(see [7]). The $B-$term generalises the usual $\frac{c}{Q_{2}^{2}}$ term.

##  4 The $(q_{i},p_{i})$ Coordinates and the Calogero-Moser Potential

We now consider these systems in the original coordinates $(q_{i},p_{i})$. Our
main interest is the Calogero-Moser Potential with $B(z)$ (from (7a)) and
$W(\zeta)$ (from (8a)) taking the specific forms

$B(z)=\frac{9g^{2}(1+z^{2})^{2}}{2(1-3z^{2})^{2}},\quad
W(\zeta)=g^{2}\left(1+\frac{1}{\zeta^{2}}+\frac{1}{(1+\zeta)^{2}}\right).$
(15)

###  4.1 Kepler-Calogero-Moser System

Rewriting the Kepler case of Section 3.3, we find the Kepler-Calogero-Moser
System [14]:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})+g^{2}\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right)-\frac{\mu}{\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}},$
(16a) $\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle(p_{1}+p_{2}+p_{3})S-2(q_{1}+q_{2}+q_{3})H-\frac{\mu(q_{1}+q_{2}+q_{3})}{\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}},$
(16b) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(J_{1}+J_{2}+J_{3})^{2}$ (16c)
$\displaystyle+4g^{2}(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}-q_{1}q_{2}-q_{2}q_{3}-q_{3}q_{1})\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right),$
$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle
2(q_{1}^{2}+q_{2}^{2}+q_{3}^{2})H-S^{2}+2\mu\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}.$
(16d)

As well as $\\{H,X_{i}\\}=0$, we have a further cubic integral $X_{3}$,
defined by $\\{X_{1},\Omega\\}=-2X_{3}$, where

$\displaystyle X_{3}$ $\displaystyle=$
$\displaystyle(p_{1}+p_{2}+p_{3})\Omega-\frac{\mu}{\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}}\,((q_{2}^{2}+q_{3}^{2}-q_{1}(q_{2}+q_{3}))p_{1}$
(16e)
$\displaystyle\qquad+(q_{3}^{2}+q_{1}^{2}-q_{2}(q_{3}+q_{1}))p_{2}+(q_{1}^{2}+q_{2}^{2}-q_{3}(q_{1}+q_{2}))p_{3}).$

The remaining nonzero Poisson relations are listed below

$\\{X_{1},X_{3}\\}=2(X_{2}-6\Omega)H+X_{1}^{2}-3\mu^{2},\quad\\{\Omega,X_{3}\\}=-2X_{1}\Omega.$
(16f)

The functions $H,X_{1},X_{2},\Omega,X_{3}$ have rank 4 and satisfy

$X_{3}^{2}=\left(2(3\Omega-
X_{2})H-X_{1}^{2}+3\mu^{2}\right)\Omega-\mu^{2}X_{2}.$ (16g)

###  4.2 A Calogero-Moser System Coupled with the Potential $U_{2}$

Rewriting the case of Equation (12a), we obtain

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})+g^{2}\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right)$
(17a) $\displaystyle\hskip
142.26378pt+\omega^{2}(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}+q_{1}q_{2}+q_{2}q_{3}+q_{3}q_{1}),$
$\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle(p_{1}+p_{2}+p_{3})S-2(q_{1}+q_{2}+q_{3})H$ (17b)
$\displaystyle\hskip
85.35826pt+\frac{4}{3}\omega^{2}(q_{1}+q_{2}+q_{3})(2q_{1}^{2}+2q_{2}^{2}+2q_{3}^{2}+q_{1}q_{2}+q_{2}q_{3}+q_{3}q_{1}),$
$\displaystyle X_{2}$ $\displaystyle=$ $\displaystyle(J_{1}+J_{2}+J_{3})^{2}$
(17c)
$\displaystyle\qquad+4g^{2}\,(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}-q_{1}q_{2}-q_{2}q_{3}-q_{3}q_{1})\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right),$
$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle(p_{1}+p_{2}+p_{3})^{2}+4\omega^{2}(q_{1}+q_{2}+q_{3})^{2}.$
(17d)

We then have $\\{X_{1},F_{1}\\}=-4F_{2}$, where

$\displaystyle F_{2}$ $\displaystyle=$ $\displaystyle
F_{2}^{(1)}-3F_{2}^{(2)}+3\omega^{2}(q_{1}+q_{2}+q_{3})(q_{1}p_{1}+q_{2}p_{2}+q_{3}p_{3})$
(18a)
$\displaystyle\qquad-3\omega^{2}\left((q_{2}^{2}+q_{3}^{2}+q_{2}q_{3})p_{1}+(q_{1}^{2}+q_{3}^{2}+q_{1}q_{3})p_{2}+(q_{1}^{2}+q_{2}^{2}+q_{1}q_{2})p_{3}\right),$
with $\displaystyle F_{2}^{(1)}$ $\displaystyle=$ $\displaystyle
p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+3g^{2}\left(\frac{p_{1}+p_{2}}{(q_{1}-q_{2})^{2}}+\frac{p_{1}+p_{3}}{(q_{1}-q_{3})^{2}}+\frac{p_{2}+p_{3}}{(q_{2}-q_{3})^{2}}\right),$
(18b) $\displaystyle F_{2}^{(2)}$ $\displaystyle=$ $\displaystyle
p_{1}p_{2}p_{3}-g^{2}\left(\frac{p_{3}}{(q_{1}-q_{2})^{2}}+\frac{p_{2}}{(q_{1}-q_{3})^{2}}+\frac{p_{1}}{(q_{2}-q_{3})^{2}}\right).$
(18c) We write it like this because, when $\omega=0$, both $F_{2}^{(1)}$ and
$F_{2}^{(2)}$ are first integrals.

The remaining nonzero brackets are

$\\{F_{1},F_{2}\\}=-36\omega^{2}X_{1},\quad\\{X_{1},F_{2}\\}=-\frac{3}{2}(6H-F_{1})(2H-F_{1})+6\omega^{2}X_{2}.$
(18d)

This algebra has rank 4, since

$F_{2}^{2}=\frac{1}{4}F_{1}(F_{1}-6H)^{2}-3\omega^{2}(F_{1}X_{2}+3X_{1}^{2}).$
(18e)

###### Remark 4.1

Writing the oscillator term as

$\omega^{2}(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}+q_{1}q_{2}+q_{2}q_{3}+q_{3}q_{1})=2\omega^{2}(q_{1}^{2}+q_{2}^{2}+q_{3}^{2})-\frac{1}{2}\,\omega^{2}((q_{1}-q_{2})^{2}+(q_{2}-q_{3})^{2}+(q_{3}-q_{1})^{2}),$
(18f)

it can be interpreted as a modification of the “nearest neighbour
interaction”, together with an external isotropic oscillator.

##  5 The Conformally Flat Case

In this and the following sections, we extend our results to the conformally
flat case. We start in the separation coordinates, $(x,y,z)$, adding a
conformal factor to the entire separable Hamiltonian (6b), but requiring that
the resulting Hamiltonian is still separable in these coordinates. The
functions $X_{1},\,X_{2}$ are thus extended to this case.

In Section 5.2, we transform these functions into the $(Q_{i},P_{i})$
coordinates of (3) and then (in Section 6) extend the results of Section 3.

In Section 7 we consider restrictions of the conformal factor by imposing
various 2 and 3 dimensional symmetry algebras on the kinetic energy.

###  5.1 In the $(x,y,z)$ Coordinates

We add a conformal factor to the entire separable Hamiltonian (6b)

$H=\varphi(x,y,z)\left(\frac{2xp_{x}^{2}+2yp_{y}^{2}+A_{1}(x)+A_{2}(y)}{x+y}+\frac{(1+z^{2})^{2}p_{z}^{2}+2(1+z^{2})B(z)}{2xy}\right),$
(19a) demanding that $H=E$ can still be separated. This gives that $\varphi$
must have the form
$\varphi=\frac{xy(x+y)}{xy(\varphi_{1}(x)+\varphi_{2}(y))+(x+y)\varphi_{3}(z)},$
(19b) and repeating the construction of $X_{1}$ and $X_{2}$, we see they are
modified as follows: $\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle\frac{2(x^{2}p_{x}^{2}-y^{2}p_{y}^{2})}{x+y}+\frac{y\varphi_{2}(y)-x\varphi_{1}(x)}{x+y}H+\frac{xA_{1}(x)-yA_{2}(y)}{x+y},$
(19c) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(1+z^{2})^{2}p_{z}^{2}+2(1+z^{2})B(z)-2\varphi_{3}(z)H,$ (19d)

with $H,\,X_{1}$ and $X_{2}$ still in involution.

###  5.2 In the $(Q_{i},P_{i})$ Coordinates

Rewriting this in the $(Q_{i},P_{i})$ coordinates, the system (19) takes the
form

$\displaystyle H$ $\displaystyle=$
$\displaystyle\varphi\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{A_{1}(x)+A_{2}(y)}{2R}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right),$
(20a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S+\frac{y\varphi_{2}(y)-x\varphi_{1}(x)}{2R}H+\frac{xA_{1}(x)-yA_{2}(y)}{2R},$
(20b) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}-2\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)H+\frac{2(Q_{2}^{2}+Q_{3}^{2})}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(20c) where
$\varphi=\frac{2(Q_{2}^{2}+Q_{3}^{2})R}{(Q_{2}^{2}+Q_{3}^{2})(\varphi_{1}(x)+\varphi_{2}(y))+2R\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)},$
(20d) with $x=R+Q_{1}$, $y=R-Q_{1}$ and
$R=\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}$.

When $\varphi_{1}(x)=x,\ \varphi_{2}(y)=y,\
\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)=0$, this system reduces to (7). In
this way, we extend the completely integrable system (7) to a conformally flat
case.

##  6 Some Specific Potentials in the Conformally Flat Case

Here we extend the cases of Section 3 to conformally flat metrics. Whilst the
restriction of the functions $A_{i}$ is allowed without any further
constraints on the conformal factor $\varphi$, demanding additional first
integrals (such as $\Omega$ in the Kepler case) does force constraints on
$\varphi$.

###  6.1 The Case $A_{i}=0$

Setting $A_{i}=0$ (as in Section 3.2) does not change the result that the
functions (20) are in involution, so is no constraint on the function
$\varphi$. However, asking for an extension of the function (10d) to commute
with $H$ does restrict $\varphi$. We omit the details, but requiring
$\\{H,\Omega\\}=0$, where $\Omega$ has the form

$\Omega=-S^{2}+\psi(Q_{1},Q_{2},Q_{3})H+V(Q_{1},Q_{2},Q_{3}),$ (21a) leads to
$\varphi_{1}=k_{1}x+\frac{k_{2}}{x},\quad\varphi_{2}=k_{1}y+\frac{k_{3}}{y}+k_{4},\quad\psi=2k_{1}(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+k_{4}\sqrt{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}},\quad
V=0,$ (21b) giving
$\varphi=\frac{2(Q_{2}^{2}+Q_{3}^{2})R}{\left(2k_{1}(Q_{2}^{2}+Q_{3}^{2})+2\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)+k_{2}+k_{3}\right)R+k_{4}(Q_{2}^{2}+Q_{3}^{2})+(k_{3}-k_{2})Q_{1}}.$
(21c)

Imposing the additional constraint $\\{H,P_{1}\\}=0$ leads to
$k_{3}-k_{2}=k_{4}=0$, giving

$H=\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right),$
(22a) with similar modifications to $X_{1},X_{2}$ and $\Omega$, which now
satisfy
$\\{P_{1},X_{1}\\}=2k_{1}H-P_{1}^{2},\quad\\{P_{1},\Omega\\}=2X_{1},\quad\\{X_{1},\Omega\\}=-2P_{1}\Omega.$
(22b) These 5 functions have rank 4, satisfying the constraint
$X_{1}^{2}=4k_{1}k_{2}H^{2}+2k_{1}H(\Omega-X_{2})-P_{1}^{2}\Omega,$ (22c)
generalising (10f).

######  Remark 6.1 (Further Symmetry Constraint)

If we set $\varphi_{3}=0$, in (22a), then the metric has an additional
rotational symmetry (with respect to $J_{1}$) and coincides with that of (4a)
in [12]. If we ask for the whole Hamiltonian to commute with this rotation,
then we obtain

$H=\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{k_{5}}{Q_{2}^{2}+Q_{3}^{2}}\right),$

which is a reduced form of (4.6a) in [13]. Such constraints are discussed
further in Section 7.

###  6.2 The Kepler Case

Here we extend the integrals (11), to obtain

$\displaystyle H$ $\displaystyle=$
$\displaystyle\varphi\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)-\frac{\mu}{R}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right),$
(23a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S+\left(-2k_{1}Q_{1}+\frac{k_{3}}{2}-\frac{k_{3}Q_{1}-k_{2}}{2R}\right)H-\frac{\mu
Q_{1}}{R},$ (23b) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}-2\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)H+\frac{2(Q_{2}^{2}+Q_{3}^{2})}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(23c) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle-S^{2}+\left(2k_{1}R^{2}+k_{3}R\right)H+2\mu R,$ (23d) with
$\varphi=\frac{2(Q_{2}^{2}+Q_{3}^{2})R}{2\left(k_{1}(Q_{2}^{2}+Q_{3}^{2})+\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)\right)R+k_{3}(Q_{2}^{2}+Q_{3}^{2})+k_{2}Q_{1}}.$
(23e)

Again, this restriction on $\varphi$ is a result of demanding the existence of
the extended $\Omega$.

###  6.3 The Oscillator Potential $U_{2}$

Again, specifying $A_{i}$ to obtain $U_{2}$ does not change the result that
the functions (20) are in involution, so is no constraint on the function
$\varphi$. However, asking for an extension of either (12d) or (13c) does lead
to restrictions. Unlike the flat case, we cannot impose both of these
simultaneously, without forcing $\varphi$ to be constant, thus returning to
the flat case.

####  6.3.1 Extending (12d)

With $F_{1}=P_{1}^{2}+\psi(Q_{1},Q_{2},Q_{3})H+V(Q_{1},Q_{2},Q_{3})$, we find
$\varphi_{1}(x)=k_{1}x+\frac{k_{2}}{x},\;\varphi_{2}(y)=k_{1}y+\frac{k_{2}}{y}$,
leading to

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}+\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{\omega^{2}}{2}(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right),$
(24a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-2k_{1}Q_{1}H+2\omega^{2}Q_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}),$ (24b)
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}-2\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)H+\frac{2(Q_{2}^{2}+Q_{3}^{2})}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(24c) $\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle
P_{1}^{2}+4\omega^{2}Q_{1}^{2}.$ (24d) These four functions generate the
Poisson algebra:
$\\{X_{1},F_{1}\\}=-2F_{2},\;\;\\{X_{1},F_{2}\\}=(3F_{1}-2k_{1}H)(2k_{1}H-F_{1})+4\omega^{2}(X_{2}-2k_{2}H),\;\;\\{F_{1},F_{2}\\}=-8\omega^{2}X_{1},$
(24e) where
$F_{2}=P_{1}(2k_{1}H-P_{1}^{2})+2\omega^{2}(2Q_{1}(Q_{2}P_{2}+Q_{3}P_{3})-(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})P_{1}).$
(24f) These 5 functions have rank 4, satisfying
$F_{2}^{2}=F_{1}(2k_{1}H-F_{1})^{2}-4\omega^{2}(X_{1}^{2}+X_{2}F_{1}-2k_{2}F_{1}H).$
(24g)

####  6.3.2 Extending (13c)

With $F_{3}^{(2)}=P_{2}^{2}+\omega^{2}Q_{2}^{2}+V_{3}^{(2)}(Q_{2})$, we find

$\displaystyle\varphi_{1}(x)=k_{3}x^{2}+k_{1}x-\frac{k_{5}}{x},\quad\varphi_{2}(y)=-k_{3}y^{2}+k_{1}y-\frac{k_{5}}{y},\quad\varphi_{3}(z)=\frac{k_{6}}{z^{2}}+k_{5}+k_{6},$
$\displaystyle B(z)=\frac{k_{2}}{2}+\frac{k_{4}}{2z^{2}},\quad
V_{3}^{(2)}(Q_{2})=\frac{k_{2}}{Q_{2}^{2}},$

leading to

$H=\frac{Q_{3}^{2}}{(2k_{3}Q_{1}+k_{1})Q_{3}^{2}+k_{6}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{\omega^{2}}{2}(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+\frac{k_{2}}{2Q_{2}^{2}}+\frac{k_{4}}{2Q_{3}^{2}}\right).$

Unlike the flat case, we must separately require the further integral

$F_{3}^{(3)}=P_{3}^{2}+\omega^{2}Q_{3}^{2}+V_{3}^{(3)}(Q_{3}),\quad\mbox{which
imposes}\quad k_{6}=0,\;V_{3}^{(3)}(Q_{2})=\frac{k_{4}}{Q_{3}^{2}},$

leading to a conformally flat version of the last case in Table 1 of [6]:

$\displaystyle
H=\frac{1}{2k_{3}Q_{1}+k_{1}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{\omega^{2}}{2}(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+\frac{k_{2}}{2Q_{2}^{2}}+\frac{k_{4}}{2Q_{3}^{2}}\right),$
(25a) $\displaystyle
X_{1}=P_{1}S-(k_{3}(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+2k_{1}Q_{1})H+2\omega^{2}Q_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}),$
(25b) $\displaystyle
X_{2}=(Q_{2}P_{3}-Q_{3}P_{2})^{2}+(Q_{2}^{2}+Q_{3}^{2})\left(\frac{k_{2}}{Q_{2}^{2}}+\frac{k_{4}}{Q_{3}^{2}}\right),$
(25c) $\displaystyle
X_{3}=P_{2}^{2}+\omega^{2}Q_{2}^{2}+\frac{k_{2}}{Q_{2}^{2}},\quad
X_{4}=P_{3}^{2}+\omega^{2}Q_{3}^{2}+\frac{k_{4}}{Q_{3}^{2}}.$ (25d)

This Hamiltonian is clearly separable and this separability is directly
related to the involutive triple $H,X_{3},X_{4}$. We also have the involutive
triple $H,X_{1},X_{2}$, which is related to separation in the $x,y,z$
coordinates.

We can add two more simple quadratic integrals:

$X_{5}=P_{2}J_{3}+k_{3}Q_{2}^{2}\,H+Q_{1}\left(\frac{k_{2}}{Q_{2}^{2}}-\omega^{2}Q_{2}^{2}\right),\qquad
X_{6}=-P_{3}J_{2}+k_{3}Q_{3}^{2}\,H+Q_{1}\left(\frac{k_{4}}{Q_{3}^{2}}-\omega^{2}Q_{3}^{2}\right),$
(26a) which satisfy $X_{1}+X_{5}+X_{6}=0$, so have essentially replaced
$X_{1}$ by two simpler integrals. These are related to integrals of 2D
resonant oscillators, as noted in (13a) for the flat case.

To build the Poisson algebra generated by these integrals, we need four
additional cubic integrals:

$X_{7}=\frac{1}{4}\\{X_{2},X_{3}\\},\quad
X_{8}=\frac{1}{2}\\{X_{2},X_{5}\\},\quad
X_{9}=\frac{1}{2}\\{X_{3},X_{5}\\},\quad X_{10}=\frac{1}{2}\\{X_{4},X_{6}\\}.$
(26b)

The remaining Poisson brackets between the quadratic elements
$X_{2},\dots,X_{6}$ are given by

$\\{X_{2},X_{3}+X_{4}\\}=\\{X_{2},X_{5}+X_{6}\\}=\\{X_{3},X_{4}\\}=\\{X_{3},X_{6}\\}=\\{X_{4},X_{5}\\}=0,\quad\\{X_{5},X_{6}\\}=X_{7}.$
(26c)

The brackets with the cubic elements are more complicated, so given in the
Appendix (Section 10.2).

The integrals $H,\,X_{2},\,X_{3},\,X_{4}$ and $X_{5}$ are functionally
independent, so this system is maximally superintegrable. The entire algebra,
including $H$, is 10 dimensional, so subject to polynomial constraints, also
given in the Appendix.

######  Remark 6.2 (Darboux-Koenigs $D_{1}$)

On the level surface $X_{4}=m_{4}$, the Hamiltonian (25a) reduces to

$H=\frac{1}{2k_{3}Q_{1}+k_{1}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}\right)+\frac{\omega^{2}}{2}(4Q_{1}^{2}+Q_{2}^{2})+\frac{k_{2}}{2Q_{2}^{2}}+\frac{m_{4}}{2}\right),$

which is the Darboux-Koenigs $D_{1}$ kinetic energy with a Case 1 potential in
the classification of [17]. Two quadratic integrals which commute with $X_{4}$
are $X_{3}$ and $X_{5}$, which are the ones listed in [17].

###  6.4 The Generalised Hénon-Heiles Potential $U_{3}$

Choosing $A_{i}$ to give the form of (14a), imposes no restrictions on
$\varphi$. However, imposing $\\{\varphi,P_{1}\\}=0$ (Killing vector of the
metric), leads to

$\varphi_{1}(x)=k_{1}x+\frac{k_{2}}{x},\quad\varphi_{2}(y)=k_{1}y+\frac{k_{2}}{y},\quad\mbox{so}\quad\varphi=\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}+\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)}.$
(27a) Now requiring the entire Hamiltonian to commute with
$J_{1}=Q_{2}P_{3}-Q_{3}P_{2}$, implies that $\varphi_{3}=\mbox{const}$
(absorbed into $k_{2}$) and $B(z)=\frac{k_{3}}{1+z^{2}}$, giving
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{\omega^{2}}{2}(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+4kQ_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+\frac{k_{3}}{Q_{2}^{2}+Q_{3}^{2}}\right),$
$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-2k_{1}Q_{1}H+2\omega^{2}Q_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})+k\left(16Q_{1}^{4}+12Q_{1}^{2}(Q_{2}^{2}+Q_{3}^{2})+(Q_{2}^{2}+Q_{3}^{2})^{2}\right),$
with $X_{2}$ reducing to $J_{1}^{2}+2k_{3}$.

In the context of [12, 13], the corresponding metric has symmetry algebra
$\left<e_{1},h_{4}\right>$ (where $h_{4}=J_{1}$), so we can adapt our
“universal coordinates” to reduce this Hamiltonian to 2 degrees of freedom.
This is discussed in Section 7.1.

##  7 The Conformally Flat Case with Symmetries

In [12, 13] we made a systematic study of subalgebras of the full conformal
algebra (in 3 dimensions). For each subalgebra in the classification, we
considered the restriction of the general conformal factor $\varphi$, in order
to be invariant under the action of the subalgebra.

Here we consider such restrictions of our conformal factor $\varphi$, given by
(20d). Our aim is not to give a full classification, but to focus on some of
the more important examples. As explained in [13], for each symmetry algebra,
there exist universal coordinates, in which the kinetic energy is in separable
form. In [13], we then added separable potentials, but here we are given a
potential by the construction. We can, however, restrict this to be invariant
with respect to one of the symmetries. The resulting Hamiltonian can be
written in terms of the corresponding universal coordinates, making one of the
coordinates “ignorable”, so the Hamiltonian can be interpreted as a 2
dimensional reduction. First integrals which commute with the adapted symmetry
can also be reduced to this 2 dimensional space.

The formulae for $S$ and $J_{i}$ are given in (58), in terms of the
coordinates $q_{i},p_{i}$. Here, of course, we need the equivalent formulae in
terms of $Q_{i},P_{i}$.

###  7.1 The 2D Algebra $\left<P_{1},J_{1}\right>$

This algebra is commutative and referred to as $\left<e_{1},h_{4}\right>$ in
[12, 13].

It is simple to check that

$\\{\varphi,P_{1}\\}=\\{\varphi,J_{1}\\}=0\quad\Rightarrow\quad\varphi=\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}},$
(28)

with
$\varphi_{1}(x)=k_{1}x+\frac{k_{2}}{x},\;\varphi_{2}(y)=k_{1}y+\frac{k_{2}}{y},\;\varphi_{3}(z)=0$.

We then consider the Hamiltonian (20a), with this particular $\varphi$.

####  7.1.1 The Case with $\\{P_{1},H\\}=0$

If, in addition, we impose $\\{P_{1},H\\}=0$, then (20a) reduces to

$H=\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{k_{4}}{Q_{2}^{2}+Q_{3}^{2}}+k_{3}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right),$
(29a) with
$A_{1}(x)=k_{3}x+\frac{k_{4}}{x},\;A_{2}(y)=k_{3}y+\frac{k_{4}}{y}$.

The canonical transformation, with

$u=Q_{1},\quad v=\frac{1}{2}\,\log\left(Q_{2}^{2}+Q_{3}^{2}\right),\quad
w=\arctan\left(\frac{Q_{2}}{Q_{3}}\right),$ (29b)

gives $P_{1}=p_{u}$ and

$H=\frac{1}{k_{1}e^{2v}+k_{2}}\,\left(\frac{1}{2}(p_{v}^{2}+p_{w}^{2})+\frac{1}{2}e^{2v}(p_{u}^{2}+2k_{3})+k_{4}+\frac{B(\cot
w)}{\sin^{2}w}\right).$ (29c)

Considered as a 2 dimensional system on the level surface
$p_{u}=\mbox{const}$, this Hamiltonian is separable, so gives a second
quadratic integral

$K_{1}=p_{w}^{2}+\frac{2B(\cot w)}{\sin^{2}w}.$ (29d)

Any integral of (29a), which commutes with $P_{1}$, can be reduced to this
space. For example $K_{1}$ could be derived from $X_{2}$.

####  7.1.2 The Case with $\\{J_{1},H\\}=0$

If, instead, we impose $\\{J_{1},H\\}=0$, then (20a) reduces to

$H=\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{A_{1}(x)+A_{2}(y)}{2R}+\frac{k_{3}}{Q_{2}^{2}+Q_{3}^{2}}\right),$
(30a) with $B(z)=\frac{k_{3}}{z^{2}+1}$. In this case
$X_{2}=J_{1}^{2}+2k_{3}$, so we can replace $X_{2}$ by $J_{1}$ for Liouville
integrability.

The canonical transformation, with

$u=Q_{1},\quad v=\sqrt{Q_{2}^{2}+Q_{3}^{2}},\quad
w=\arctan\left(\frac{Q_{2}}{Q_{3}}\right),$ (30b)

gives $J_{1}=-p_{w}$ and

$H=\frac{v^{2}}{k_{1}v^{2}+k_{2}}\,\left(\frac{1}{2}(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2v^{2}}+\frac{A_{1}(x)+A_{2}(y)}{2\sqrt{u^{2}+v^{2}}}\right),$
(30c)

where $x=\sqrt{u^{2}+v^{2}}+u,\,y=\sqrt{u^{2}+v^{2}}-u$.

Considered as a 2 dimensional system on the level surface
$p_{w}=\mbox{const}$, this Hamiltonian is not generally separable, but can be
for particular choices of $A_{i}$, in which case it would again give a second
quadratic integral.

Any integral of (30a), which commutes with $J_{1}$, can be reduced to this
space.

####  7.1.3 The Kepler Case (23a) with this $\varphi$

It can be seen that setting $k_{2}=k_{3}=0$ and
$\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)=\bar{k}_{2}$ (but then dropping
the bar), that the conformal factor of (23a) reduces to (28), giving the
Hamiltonian:

$H=\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)-\frac{\mu}{R}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right).$
(31a) The remaining integrals of (23) can similarly be reduced: $\displaystyle
X_{1}$ $\displaystyle=$ $\displaystyle P_{1}S-2k_{1}Q_{1}H-\frac{\mu
Q_{1}}{R},$ (31b) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}-2k_{2}H+\frac{2(Q_{2}^{2}+Q_{3}^{2})}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(31c) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle-S^{2}+2k_{1}R^{2}H+2\mu R,$ (31d)

If we restrict the Hamiltonian (31a) further, by demanding that
$\\{J_{1},H\\}=0$ (as in (30a)), then we find that

$H=\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)-\frac{\mu}{R}+\frac{k_{3}}{Q_{2}^{2}+Q_{3}^{2}}\right),$
(32)

and similarly for $X_{1}$ and $\Omega$, since
$\\{J_{1},X_{1}\\}=\\{J_{1},\Omega\\}=0$. We find that $X_{2}$ can be replaced
by $J_{1}$ itself.

If we now use the canonical transformation (30b), we obtain $J_{1}=-p_{w}$ and

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{k_{1}v^{2}+k_{2}}\left(\frac{1}{2}(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2v^{2}}-\frac{\mu}{\sqrt{u^{2}+v^{2}}}\right),$
(33a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
p_{u}(up_{u}+vp_{v})-2k_{1}uH-\frac{\mu\,u}{\sqrt{u^{2}+v^{2}}},$ (33b)
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle(up_{v}-vp_{u})^{2}-\frac{2k_{2}(u^{2}+v^{2})}{v^{2}}\,H+\frac{(2k_{3}+p_{w}^{2})(u^{2}+v^{2})}{v^{2}}.$
(33c)

######  Remark 7.1 (Kepler in $D_{2}$ background)

This Hamiltonian has a Darboux-Koernigs $D_{2}$ kinetic energy, with 2
independent quadratic integrals, but not separable in these coordinates. It is
a superintegrable Kepler system in a $D_{2}$ geometry. This is a deformation
of the Kepler problem in 2D, with $X_{1}$ a Runge-Lenz integral, whilst
$X_{2}$ is a deformation of rotational invariance.

####  7.1.4 The Hamiltonian (30a) with the Generalised Hénon-Heiles Potential

Here we consider the Hamiltonian (LABEL:U2U3HX1), which is a particular case
of (30a). The canonical transformation (30b) gives

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{k_{1}v^{2}+k_{2}}\left(\frac{1}{2}(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2v^{2}}+\frac{\omega^{2}}{2}(4u^{2}+v^{2})+4ku(2u^{2}+v^{2})\right),$
(34a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
p_{u}(up_{u}+vp_{v})-2k_{1}uH+2\omega^{2}u(2u^{2}+v^{2})+k(16u^{4}+12u^{2}v^{2}+v^{4}),$
(34b)

in which $p_{w}=-J_{1}$, a first integral. On each level surface
$p_{w}=\mbox{const}$, this represents an integrable system in 2 degrees of
freedom. If $k_{2}=0$, this reduces to the standard “KdV case” of integrable
Hénon-Heiles models (see [7]). When $k_{1}=0$, this is an extension to a
constant curvature space. For generic $k_{1}k_{2}\neq 0$, the metric is of
Darboux-Koenigs type $D_{2}$. When $k=0$ this potential is of “type A” in the
classification of [16] and the system is actually super-integrable, but the
Hénon-Heiles case, with $k\neq 0$ is just integrable.

####  7.1.5 The Hamiltonian (30a) with Potential $U_{4}$

If we replace $U_{3}$ in (LABEL:U2U3HX1) by $U_{4}$, we obtain another
interesting subcase of (30a).

Specifically, if we set
$A_{1}(x)=\frac{1}{2}\omega^{2}x^{3}+kx^{5},\,A_{2}(y)=\frac{1}{2}\omega^{2}y^{3}+ky^{5}$,
then

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{Q_{2}^{2}+Q_{3}^{2}}{k_{1}(Q_{2}^{2}+Q_{3}^{2})+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{\omega^{2}}{2}(4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})\right.$
(35a) $\displaystyle\hskip
142.26378pt\left.+k(16Q_{1}^{4}+12Q_{1}^{2}(Q_{2}^{2}+Q_{3}^{2})+(Q_{2}^{2}+Q_{3}^{2})^{2})+\frac{k_{3}}{Q_{2}^{2}+Q_{3}^{2}}\right),$
$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-2k_{1}Q_{1}H+2\omega^{2}Q_{1}(2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})$ (35b)
$\displaystyle\hskip
170.71652pt+2kQ_{1}\left(16Q_{1}^{4}+16Q_{1}^{2}(Q_{2}^{2}+Q_{3}^{2})+3(Q_{2}^{2}+Q_{3}^{2})^{2}\right),$
The canonical transformation (30b) gives $\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{k_{1}v^{2}+k_{2}}\left(\frac{1}{2}(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2v^{2}}+\frac{\omega^{2}}{2}(4u^{2}+v^{2})+k(16u^{4}+12u^{2}v^{2}+v^{4})\right),$
(35c) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
p_{u}(up_{u}+vp_{v})-2k_{1}uH+2\omega^{2}u(2u^{2}+v^{2})+2ku(16u^{4}+16u^{2}v^{2}+3v^{4}),$
(35d) in which $p_{w}=-J_{1}$, a first integral.

Again, as a 2 dimensional sytem and for generic $k_{1}k_{2}\neq 0$, the metric
is of Darboux-Koenigs type $D_{2}$. When $k=0$ it is a “type A” potential in
the classification of [16]. The only difference between (34a) and (35c) is
that the Hénon-Heiles part of the potential has been replaced by the quartic
potential in the 2 dimensional “parabolic” series.

###  7.2 The 2D Algebra $\left<S,J_{1}\right>$

This algebra is commutative and would be referred to as
$\left<h_{1},h_{4}\right>$ in [12, 13], which, in that context, is equivalent
to $\left<h_{1},h_{2}\right>$ in our classification. However, it is clear from
the formulae (20) that $J_{1}$ ($=h_{4}$) acts in a special way, so this
equivalence no longer holds.

Since $S$ acts conformally on
$H^{0}=\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})$, we need to consider
$\varphi H^{0}$ instead of $\varphi$:

$\\{\varphi H^{0},S\\}=\\{\varphi
H^{0},J_{1}\\}=0\quad\Rightarrow\quad\varphi=\frac{(Q_{2}^{2}+Q_{3}^{2})R}{k_{1}R+k_{2}Q_{1}},$
(36)

where
$k_{1}=\frac{1}{2}(\bar{k}_{1}+\bar{k}_{2}+2\bar{k}_{3}),\;k_{2}=\frac{1}{2}(\bar{k}_{2}-\bar{k}_{1})$
and
$\varphi_{1}=\frac{\bar{k}_{1}}{x},\;\varphi_{2}=\frac{\bar{k}_{2}}{y},\;\varphi_{3}(z)=\bar{k}_{3}$.

####  7.2.1 The Case with $\\{S,H\\}=0$

If, in addition, we impose $\\{S,H\\}=0$, then (20a) reduces to

$H=\frac{(Q_{2}^{2}+Q_{3}^{2})R}{k_{1}R+k_{2}Q_{1}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{k_{3}+k_{4}}{2(Q_{2}^{2}+Q_{3}^{2})}+\frac{(k_{4}-k_{3})Q_{1}}{2(Q_{2}^{2}+Q_{3}^{2})R}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right),$
(37a) with $A_{1}(x)=\frac{k_{3}}{x},\;A_{2}(y)=\frac{k_{4}}{y}$.

The canonical transformation, with

$u=\frac{1}{2}\,\log\left(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\right),\quad
v=\log\left(\frac{Q_{1}+R}{\sqrt{Q_{2}^{2}+Q_{3}^{2}}}\right),\quad
w=\arctan\left(\frac{Q_{2}}{Q_{3}}\right),$ (37b)

gives $S=p_{u}$ and

$H=\frac{1}{k_{1}+k_{2}\tanh
v}\,\left(\frac{1}{2}(p_{v}^{2}+p_{w}^{2})+\frac{p_{u}^{2}}{2\cosh^{2}v}+\frac{1}{2}(k_{3}+k_{4})+\frac{1}{2}(k_{4}-k_{3})\tanh
v+\frac{B(\cot w)}{\sin^{2}w}\right).$ (37c)

Considered as a 2 dimensional system on the level surface
$p_{u}=\mbox{const}$, this Hamiltonian is separable, so gives a second
quadratic integral

$K_{1}=p_{w}^{2}+\frac{2B(\cot w)}{\sin^{2}w}.$ (37d)

Any integral of (37a), which commutes with $S$, can be reduced to this space.
For example $K_{1}$ could be derived from $X_{2}$.

####  7.2.2 The Case with $\\{J_{1},H\\}=0$

If, instead, we impose $\\{J_{1},H\\}=0$, then (20a) reduces to

$H=\frac{(Q_{2}^{2}+Q_{3}^{2})R}{k_{1}R+k_{2}Q_{1}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{A_{1}(x)+A_{2}(y)}{2R}+\frac{k_{3}}{Q_{2}^{2}+Q_{3}^{2}}\right),$
(38a) with $B(z)=\frac{k_{3}}{z^{2}+1}$.

The canonical transformation, with

$u=\frac{1}{2}\,\log\left(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\right),\quad
v=\arctan\left(\frac{Q_{1}}{\sqrt{Q_{2}^{2}+Q_{3}^{2}}}\right),\quad
w=\arctan\left(\frac{Q_{2}}{Q_{3}}\right),$ (38b)

gives $J_{1}=-p_{w}$ and

$H=\frac{\cos^{2}v}{k_{1}+k_{2}\sin
v}\,\left(\frac{1}{2}(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2\cos^{2}v}+\frac{1}{2}e^{u}(A_{1}(x)+A_{2}(y))\right),$
(38c)

where $x=e^{u}(1+\sin v),\,y=e^{u}(1-\sin v)$

Considered as a 2 dimensional system on the level surface
$p_{w}=\mbox{const}$, this Hamiltonian is not generally separable, but can be
for particular choices of $A_{i}$, in which case it would again give a second
quadratic integral.

Any integral of (38a), which commutes with $J_{1}$, can be reduced to this
space.

####  7.2.3 The Kepler Case (23a) with this $\varphi$

It can be seen that setting $k_{1}=k_{3}=0$ and
$\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)=\bar{k}_{1},\,k_{2}=\bar{k}_{2}$
in the conformal factor (23e) (but dropping the bars), then (23a) reduces to
(36), giving the Hamiltonian:

$H=\frac{(Q_{2}^{2}+Q_{3}^{2})R}{k_{1}R+k_{2}Q_{1}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)-\frac{\mu}{R}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right).$
(39a) The remaining integrals of (23) can similarly be reduced: $\displaystyle
X_{1}$ $\displaystyle=$ $\displaystyle P_{1}S+k_{2}\,\frac{H}{R}-\frac{\mu
Q_{1}}{R},$ (39b) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}+\frac{2(Q_{2}^{2}+Q_{3}^{2})}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(39c) $\displaystyle\Omega$ $\displaystyle=$ $\displaystyle-S^{2}+2\mu R,$
(39d)

If we restrict the Hamiltonian (39a) further, by demanding that
$\\{J_{1},H\\}=0$ (as in (38a)), then we find that

$H=\frac{(Q_{2}^{2}+Q_{3}^{2})R}{k_{1}R+k_{2}Q_{1}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)-\frac{\mu}{R}+\frac{k_{3}}{Q_{2}^{2}+Q_{3}^{2}}\right),$
(40)

and similarly for $X_{1}$ and $\Omega$. We find that $X_{2}$ can be replaced
by $J_{1}$ itself.

The canonical transformation (38b) gives the following system in 2 dimensions
(for $p_{w}$ a constant)

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{\cos^{2}v}{k_{1}+k_{2}\sin
v}\,\left(\frac{1}{2}(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2\cos^{2}v}-\mu
e^{u}\right),$ (41a) $\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
e^{-u}\,p_{u}\left(p_{u}\sin v+p_{v}\cos v\right)+k_{2}e^{-u}\,H-\mu\sin v,$
(41b) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle-(p_{u}^{2}-2\mu\,e^{u}).$ (41c) In these coordinate, the
Hamiltonian is separable, so gives a second quadratic integral, which is just
$-\Omega$.

For generic $k_{i}$, the Hamiltonian (41a) has a Darboux-Koenigs type $D_{4}$
kinetic energy, with a potential of “type A” in the classification of [16]. It
reduces to a constant curvature space, when $k_{2}=0$.

######  Remark 7.2 (Other Potentials)

In the $Q_{i}$ coordinates, any of the potentials considered in Section 7.1
can be written in the case of (38a). However, the canonical transformation
(38b) is based on the symmetry algebra of the kinetic energy, so only special
potentials will be well adapted to these. In [13] we concentrated on separable
potentials, but as seen in (33a), there are other potentials which give simple
and interesting forms.

###  7.3 A Copy of the 3D Algebra $\mathfrak{sl}(2)$

Here we consider a particular realisation of $\mathfrak{sl}(2)$, within the
conformal algebra, which (in [12, 13]) we label
$\left<e_{1},h_{1},f_{1}\right>$:

$\left.\begin{array}[]{l}e_{1}=P_{1},\;\;h_{1}=S,\\\\[5.69054pt]
f_{1}=(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2})P_{1}-2Q_{1}S\end{array}\right\\}\quad\Rightarrow\quad\\{h_{1},e_{1}\\}=e_{1},\;\;\\{h_{1},f_{1}\\}=-f_{1},\;\;\\{e_{1},f_{1}\\}=2h_{1}.$
(42)

It is simple to check that

$\\{\varphi H^{0},P_{1}\\}=\\{\varphi
H^{0},S\\}=0\quad\Rightarrow\quad\varphi=\frac{Q_{2}^{2}+Q_{3}^{2}}{\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)+k_{1}},$
(43)

where $\varphi_{1}=\frac{k_{1}}{x},\;\varphi_{2}=\frac{k_{1}}{y}$. It is then
simple to check that $\\{\varphi H^{0},f_{1}\\}=0$, without further
constraint.

####  7.3.1 The Case with $\\{P_{1},H\\}=0$

If, in addition, we impose $\\{P_{1},H\\}=0$, then (20a) reduces to

$H=\frac{(Q_{2}^{2}+Q_{3}^{2})}{\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)+k_{1}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{k_{3}}{Q_{2}^{2}+Q_{3}^{2}}+k_{2}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right),$
(44a) with
$A_{1}(x)=k_{2}x+\frac{k_{3}}{x},\;A_{2}(y)=k_{2}y+\frac{k_{3}}{y}$.

The canonical transformation, with

$u=Q_{1},\quad v=\arctan\left(\frac{Q_{3}}{Q_{2}}\right),\quad
w=\frac{1}{2}\,\log(Q_{2}^{2}+Q_{3}^{2}),$ (44b)

gives $P_{1}=p_{u}$ and

$H=\frac{1}{k_{1}+\varphi_{3}(\tan
v)}\,\left(\frac{1}{2}(p_{v}^{2}+p_{w}^{2})+\frac{1}{2}\,e^{2w}(p_{u}^{2}+2k_{2})+k_{3}+\frac{B(\tan
v)}{\cos^{2}v}\right).$ (44c)

Considered as a 2 dimensional system on the level surface
$p_{u}=\mbox{const}$, this Hamiltonian is separable, so gives a second
quadratic integral

$K_{1}=p_{w}^{2}+e^{2w}(p_{u}^{2}+2k_{2}).$ (44d)

Any integral of (44a), which commutes with $P_{1}$, can be reduced to this
space. For example $K_{1}$ could be derived from $2k_{1}H-X_{2}-2k_{3}$.

####  7.3.2 The Case with $\\{S,H\\}=0$

If, instead, we impose $\\{S,H\\}=0$, then (20a) reduces to

$H=\frac{(Q_{2}^{2}+Q_{3}^{2})}{\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)+k_{1}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{k_{3}+k_{2}}{2(Q_{2}^{2}+Q_{3}^{2})}+\frac{(k_{3}-k_{2})Q_{1}}{2(Q_{2}^{2}+Q_{3}^{2})R}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right),$
(45a) with $A_{1}(x)=\frac{k_{2}}{x},\;A_{2}(y)=\frac{k_{3}}{y}$.

The canonical transformation, with

$u=\frac{1}{2}\,\log\left(\frac{R+Q_{1}}{R-Q_{1}}\right),\quad
v=\arctan\left(\frac{Q_{3}}{Q_{2}}\right),\quad
w=\frac{1}{2}\,\log(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}),$ (45b)

gives $S=p_{w}$ and

$H=\frac{1}{k_{1}+\varphi_{3}(\tan
v)}\,\left(\frac{1}{2}(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}}{2\cosh^{2}u}+\frac{1}{2}(k_{3}+k_{2})+\frac{1}{2}(k_{3}-k_{2})\tanh
u+\frac{B(\tan v)}{\cos^{2}v}\right).$ (45c)

Considered as a 2 dimensional system on the level surface
$p_{w}=\mbox{const}$, this Hamiltonian is separable, so gives a second
quadratic integral

$K_{1}=p_{u}^{2}+\frac{p_{w}^{2}}{\cosh^{2}u}+(k_{3}-k_{2})\tanh u.$ (45d)

Any integral of (45a), which commutes with $S$, can be reduced to this space.
For example $K_{1}$ could be derived from $2k_{1}H-X_{2}-k_{2}-k_{3}$.

######  Remark 7.3 (Lie algebra involution)

The case of $\\{f_{1},H\\}=0$ looks more complicated, but is related to the
$\\{P_{1},H\\}=0$ case through an involution of $\mathfrak{sl}(2)$, realised
by

$(Q_{1},Q_{2},Q_{3})\mapsto\left(\frac{Q_{1}}{R^{2}},\frac{Q_{2}}{R^{2}},\frac{Q_{3}}{R^{2}}\right)\quad\Rightarrow\quad
e_{1}\leftrightarrow f_{1},\;\;h_{1}\mapsto-h_{1}.$

###  7.4 The 3D Algebra $\mathfrak{so}(3)$

Here we consider the rotational symmetries of
$H^{0}=\frac{1}{2}(P_{1}^{2}+P_{2}^{2}+P_{3}^{2})$. In [12, 13] this algebra
is called $\left<h_{2},h_{3},h_{4}\right>$.

It is simple to check that

$\\{\varphi,J_{1}\\}=\\{\varphi,J_{2}\\}=\\{\varphi,J_{3}\\}=0\quad\Rightarrow\quad\varphi=\frac{R}{k_{1}R+k_{2}},$
(46)

with $\varphi_{1}=k_{1}x+k_{2},\;\varphi_{2}=k_{1}y+k_{2},\;\varphi_{3}=0$.

####  7.4.1 The Case with $\\{J_{1},H\\}=0$

If, in addition, we impose $\\{J_{1},H\\}=0$, then (20a) reduces to

$H=\frac{R}{k_{1}R+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+\frac{A_{1}(x)+A_{2}(y)}{2R}+\frac{k_{3}}{Q_{2}^{2}+Q_{3}^{2}}\right),$
(47a) with $B(z)=\frac{k_{3}}{z^{2}+1}$. With these values of $\varphi_{i}$
and $A_{i}$, the functions $H,X_{1},X_{2}$ are still in involution. However,
since $X_{2}=J_{1}^{2}+2k_{3}$, we can just replace it by $J_{1}$.

The canonical transformation, with

$u=\arctan\left(\frac{\sqrt{Q_{2}^{2}+Q_{3}^{2}}}{Q_{1}}\right),\quad
v=\frac{1}{2}\,\log\left(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\right),\quad
w=\arctan\left(\frac{Q_{2}}{Q_{3}}\right),$ (47b)

adapted to $J_{1}$, with $p_{w}=-J_{1}$, gives the Hamiltonian

$H=\frac{e^{-v}}{k_{1}e^{v}+k_{2}}\left(\frac{1}{2}\,(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2\sin^{2}u}+\frac{1}{2}\,e^{v}(A_{1}(x)+A_{2}(y))\right),$
(47c)

where $x=e^{v}(1+\cos u),\,y=e^{v}(1-\cos u)$.

Considered as a 2 dimensional system on the level surface
$p_{w}=\mbox{const}$, this Hamiltonian is not generally separable, but can be
for particular choices of $A_{i}$, in which case it would again give a second
quadratic integral.

Any integral of (47a), which commutes with $J_{1}$, can be reduced to this
space.

####  7.4.2 The Case with $\\{J_{2},H\\}=0$

If, instead, we impose $\\{J_{2},H\\}=0$, then (20a) reduces to

$H=\frac{R}{k_{1}R+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+k_{3}+\frac{k_{4}}{2R}+\frac{k_{5}}{Q_{2}^{2}}\right),$
(48a) with $A_{1}(x)=k_{3}x,\quad A_{2}(y)=k_{3}y+k_{4},\quad B(z)=k_{5}$.

The canonical transformation, with

$u=\arctan\left(\frac{\sqrt{Q_{1}^{2}+Q_{3}^{2}}}{Q_{2}}\right),\quad
v=\frac{1}{2}\,\log\left(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\right),\quad
w=\arctan\left(\frac{Q_{3}}{Q_{1}}\right),$ (48b)

adapted to $J_{2}$, with $p_{w}=-J_{2}$, gives a separable form of (48a):

$H=\frac{e^{-v}}{k_{1}e^{v}+k_{2}}\left(\frac{1}{2}\,(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}}{2\sin^{2}u}+k_{3}e^{2v}+\frac{1}{2}\,k_{4}e^{v}+k_{5}\sec^{2}u\right),$
(48c)

which is equivalent (on the level surface $p_{w}=\mbox{const}$) to the
Darboux-Koenigs $D_{3}$ kinetic energy, with a potential. In this separable
form, it can be seen that there is a quadratic integral

$K_{1}=p_{u}^{2}+\frac{p_{w}^{2}}{\sin^{2}u}+2k_{5}\sec^{2}u.$ (48d)

Any integral of (48a) that commutes with $J_{2}$ can be reduced to this space.
In particular, since $p_{u}^{2}+\frac{p_{w}^{2}}{\sin^{2}u}=J^{2}$, $K_{1}$ is
related to a deformation of the rotational Casimir, which is an additional
integral in this case.

######  Remark 7.4 (The Case with $\\{J_{3},H\\}=0$)

Similarly, if we impose $\\{J_{3},H\\}=0$, then (20a) reduces to

$H=\frac{R}{k_{1}R+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)+k_{3}+\frac{k_{4}}{2R}+\frac{k_{5}}{Q_{3}^{2}}\right),$

with $A_{1}(x)=k_{3}x,\quad A_{2}(y)=k_{3}y+k_{4},\quad
B(z)=\frac{k_{5}}{z^{2}}$.

####  7.4.3 Separable Case of (47c)

For general $A_{1}(x)$ and $A_{2}(y)$, there will be no additional first
integrals of (47a), but special choices can render it superintegrable.

First, we consider separability of (47c), which requires

$A_{1}(x)+A_{2}(y)=a(v)+e^{-v}b(u),$ (49a) for some functions $a(v),\,b(u)$.
If we differentiate (49a) with respect to $u$ and also with respect to $v$, we
obtain simultaneous equations for $A_{i}^{\prime}$, giving (for $A_{1}(x)$)
$2e^{2v}A_{1}^{\prime}(x)=e^{v}a^{\prime}(v)+b(u)+\tan\left(\frac{u}{2}\right)\,b^{\prime}(u)\quad\Rightarrow\quad\left(2e^{2v}A_{1}^{\prime}(x)\right)_{uv}=0.$
(49b) This, together with a similar equation for $A_{2}(y)$, give
$xA_{1xxx}+3A_{1xx}=0,\;\;\;yA_{2yyy}+3A_{2yy}=0\quad\Rightarrow\quad
A_{1}=a_{0}+a_{1}x+\frac{a_{2}}{x},\;\;A_{2}=b_{0}+b_{1}y+\frac{b_{2}}{y}.$
(49c) Substituting these back into (49a) requires one condition
($b_{1}=a_{1}$) on these 6 parameters, giving
$\frac{1}{2}\,e^{v}(A_{1}(x)+A_{2}(y))=\frac{1}{2}(a_{0}+b_{0})e^{v}+a_{1}e^{2v}+\frac{a_{2}}{2(1+\cos
u)}+\frac{b_{2}}{2(1-\cos u)}.$ (49d)

The $e^{2v}$ term can be absorbed into the $e^{v}$ term by noting that

$a_{1}e^{2v}=\frac{a_{1}}{k_{1}}\,k_{1}e^{2v}\quad\mbox{and}\quad\left(\frac{e^{-v}}{k_{1}e^{v}+k_{2}}\right)\left(\frac{a_{1}}{k_{1}}\left(k_{1}e^{2v}+k_{2}e^{v}\right)\right)=\frac{a_{1}}{k_{1}},$

so, without loss of generality, we can set $a_{1}=0$ and

$\frac{1}{2}\,e^{v}(A_{1}(x)+A_{2}(y))=\frac{1}{2}(a_{0}+b_{0})e^{v}+\frac{1}{2}(b_{2}-a_{2})\,\frac{\cos
u}{\sin^{2}u}+\frac{a_{2}+b_{2}}{2\sin^{2}u}.$ (50a) The last term can be
absorbed into the $\frac{k_{3}}{\sin^{2}u}$ term of (47c), so, defining
$\beta$ and $\gamma$ by $a_{0}+b_{0}=2\beta,\quad b_{2}-a_{2}=2\gamma,$ this
leads to
$H=\frac{e^{-v}}{k_{1}e^{v}+k_{2}}\left(\frac{1}{2}\,(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2\sin^{2}u}+\beta
e^{v}+\gamma\,\frac{\cos u}{\sin^{2}u}\right).$ (50b)

From separability, we see that

$K_{1}=p_{u}^{2}+\frac{p_{w}^{2}+2k_{3}}{\sin^{2}u}+2\gamma\,\frac{\cos
u}{\sin^{2}u}$ (50c)

is a first integral. Since $p_{u}^{2}+\frac{p_{w}^{2}}{\sin^{2}u}$ corresponds
to the rotational Casimir $J^{2}$, $K_{1}$ is an additional integral.

We also have that $\\{H,X_{1}\\}=0$ (in $Q_{1}$ coordinates), so $X_{1}$ can
be reduced to $u-v$ coordinates:

$X_{1}=e^{-v}p_{v}\left(p_{v}\cos u-p_{u}\sin u\right)-(2k_{1}e^{v}+k_{2})\cos
u\,H+\beta\cos u-\gamma e^{-v}.$ (50d)

The Hamiltonian (50b) is the Darboux-Koenigs $D_{3}$ kinetic energy, with
potential of ‘type B’ in the classification of [16], so not just separable,
but maximally superintegrable.

####  7.4.4 Superintegrability of (48a)

With the choice of $\varphi_{i},A_{i}$ and $B$ for this case, we find
(removing an additive constant from $X_{1}$)

$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-\frac{Q_{1}(k_{2}+2k_{1}R)}{R}\,H+Q_{1}\left(2k_{3}+\frac{k_{4}}{2R}\right),$
(51a) $\displaystyle X_{2}$ $\displaystyle=$ $\displaystyle
J_{1}^{2}+\frac{2k_{5}(Q_{2}^{2}+Q_{3}^{2})}{Q_{2}^{2}},$ (51b) with
$\\{X_{1},X_{2}\\}=0$ and $\displaystyle X_{3}$ $\displaystyle=$
$\displaystyle\\{X_{1},J_{2}\\}=P_{3}S-Q_{3}\left(\frac{k_{2}+2k_{1}R}{R}\right)\,H+Q_{3}\left(2k_{3}+\frac{k_{4}}{2R}\right),$
(51c) $\displaystyle X_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\\{X_{2},J_{2}\\}=J_{1}J_{3}-2k_{5}\frac{Q_{1}Q_{3}}{Q_{2}^{2}}.$
(51d)

It is easy to check that $H,X_{1},X_{2},J_{2}$, together with either of the
quantities $\\{X_{i},J_{2}\\}$, have rank 5, so this is a maximally
superintegrable system. It is a conformally flat extension of the second
system in Table I of [6].

Whilst we do not present the full Poisson algebra, there are some further
integrals, which are useful below:

$\displaystyle X_{5}$ $\displaystyle=$
$\displaystyle\\{X_{4},J_{2}\\}=J_{3}^{2}-J_{1}^{2}+2k_{5}\left(\frac{Q_{1}^{2}-Q_{3}^{2}}{Q_{2}^{2}}\right),$
(52a) $\displaystyle X_{6}$ $\displaystyle=$ $\displaystyle
J^{2}+2k_{6}\left(\frac{Q_{1}^{2}+Q_{3}^{2}}{Q_{2}^{2}}\right),$ (52b) with
$\\{X_{6},J_{2}\\}=0$. These also satisfy
$\displaystyle\\{X_{3},J_{2}\\}=-X_{1}$ $\displaystyle\Rightarrow$
$\displaystyle\\{F_{1},J_{2}\\}=0,\;\;\;\mbox{where}\;\;\;F_{1}=X_{1}^{2}+X_{3}^{2},$
(52c) $\displaystyle\\{X_{5},J_{2}\\}=-4X_{4}$ $\displaystyle\Rightarrow$
$\displaystyle\\{F_{2},J_{2}\\}=0,\;\;\;\mbox{where}\;\;\;F_{2}=4X_{4}^{2}+X_{5}^{2},$
(52d) so $F_{1}$ and $F_{2}$ are quartic integrals which commute with $J_{2}$.

#### 2D Reduction

The canonical transformation (48b) transforms (48a) into the form (48c), with
quadratic integral $K_{1}$, given by (48d). As previously said, any integral
of (48a) that commutes with $J_{2}$ can be reduced to this space. In fact
$X_{6}=K_{1}-2k_{5}$.

Whilst this potential is very similar to ‘type B’ in the classification of
[16], it has significant differences and it can be shown that there are no
further quadratic integrals of (48c). However, we have the two quartic
integrals $F_{1}$ and $F_{2}$. In fact, $F_{2}$ is related to $K_{1}$, in that
$F_{2}=(K_{1}-p_{w}^{2})^{2}-4k_{5}(K_{1}+p_{w}^{2}-k_{5})$, where $p_{w}$ is
just a parameter in this context. However, $H,K_{1},F_{1}$ are functionally
independent, so this 2D system is maximally superintegrable, but not in the
class studied in [16].

####  7.4.5 The Kepler Case (23a) with this $\varphi$

It can be seen that setting
$k_{2}=0,\,\varphi_{3}\left(\frac{Q_{3}}{Q_{2}}\right)=0$ and
$k_{3}=2\bar{k}_{2}$ in the conformal factor (23e) (but dropping the bar),
then (23e) reduces to (46), giving the Hamiltonian:

$H=\frac{R}{k_{1}R+k_{2}}\left(\frac{1}{2}\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)-\frac{\mu}{R}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)\right).$
(53a) The remaining integrals of (23) can similarly be reduced: $\displaystyle
X_{1}$ $\displaystyle=$ $\displaystyle
P_{1}S-\frac{Q_{1}(2k_{1}R+k_{2})}{R}\,H-\frac{\mu Q_{1}}{R},$ (53b)
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(Q_{2}P_{3}-Q_{3}P_{2})^{2}+\frac{2(Q_{2}^{2}+Q_{3}^{2})}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(53c) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle-S^{2}+2(k_{1}R^{2}+k_{2}R)H+2\mu R.$ (53d)

##### For $\\{J_{1},H\\}=0$

we have
$B\left(\frac{Q_{3}}{Q_{2}}\right)=\frac{k_{3}Q_{2}^{2}}{Q_{2}^{2}+Q_{3}^{2}}$,
reducing (53a) to a particular case of (47a), which reduces to a particular
example of the separable case, (50b), with $a_{2}=\gamma=0$:

$H=\frac{e^{-v}}{k_{1}e^{v}+k_{2}}\left(\frac{1}{2}\,(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}+2k_{3}}{2\sin^{2}u}-\mu
e^{v}\right),$ (54)

with corresponding particular case of $K_{1}$ and $X_{1}$. The function
$K_{1}$ just corresponds to $\Omega$ in these coordinates.

##### For $\\{J_{2},H\\}=0$

we have $B\left(\frac{Q_{3}}{Q_{2}}\right)=k_{3}$, reducing (53a) to a
particular case of (48a), and hence reduces to a particular case of (48c):

$H=\frac{e^{-v}}{k_{1}e^{v}+k_{2}}\left(\frac{1}{2}\,(p_{u}^{2}+p_{v}^{2})+\frac{p_{w}^{2}}{2\sin^{2}u}-\mu
e^{v}+k_{3}\sec^{2}u\right),$ (55)

with corresponding particular case of $K_{1}$, which corresponds to $\Omega$
in these coordinates. As we found in Section 7.4.4, $X_{1}$ does not reduce to
these coordinates and, indeed, that there are no further quadratic integrals.

However, we do have the quartic integral $F_{1}$ and, as previously remarked,
$H,K_{1},F_{1}$ are functionally independent, so this 2D system is maximally
superintegrable, but not in the class studied in [16].

##  8 A Conformally Flat Kepler-Calogero-Moser System

In Section 6.2 we presented the general conformally flat version of the Kepler
case in $(Q_{i},P_{i})$ coordinates. This involved the arbitrary function
$B\left(\frac{Q_{3}}{Q_{2}}\right)$, which, in the particular Calogero-Moser
case, should take the form $B(z)=\frac{9g^{2}(1+z^{2})^{2}}{2(1-3z^{2})^{2}}$,
as already discussed in the flat case of Section 4.1. Here we just present the
case with a spherically symmetric kinetic energy, given in Section 7.4.5:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{r}{k_{1}r+k_{2}}\left(\frac{1}{2}\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)+g^{2}\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right)-\frac{\mu}{r}\right),$
(56a) $\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle(p_{1}+p_{2}+p_{3})S-\frac{1}{r}\left((q_{1}+q_{2}+q_{3})(2k_{1}r+k_{2})\,H+\mu(q_{1}+q_{2}+q_{3})\right),$
(56b) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle(J_{1}+J_{2}+J_{3})^{2}$ (56c)
$\displaystyle+4g^{2}(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}-q_{1}q_{2}-q_{2}q_{3}-q_{1}q_{3})\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right),$
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle-S^{2}+2r\left(k_{1}r+k_{2}\right)H+2\mu\,r,$ (56d) where
$r=\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}$. As before, we need to add a cubic
integral, $\\{X_{1},\Omega\\}=-2X_{3}$, to this list, given by: $\displaystyle
X_{3}$ $\displaystyle=$
$\displaystyle(p_{1}+p_{2}+p_{3})\Omega-\frac{\mu+k_{2}H}{\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}}\,((q_{2}^{2}+q_{3}^{2}-q_{1}(q_{2}+q_{3}))p_{1}$
(56e)
$\displaystyle\qquad+(q_{3}^{2}+q_{1}^{2}-q_{2}(q_{3}+q_{1}))p_{2}+(q_{1}^{2}+q_{2}^{2}-q_{3}(q_{1}+q_{2}))p_{3}).$

These functions satisfy the Poisson relations:

$\\{X_{1},X_{3}\\}=(2k_{1}(X_{2}-6\Omega)-3k_{2}^{2}H)H+X_{1}^{2}-6\mu
k_{2}H-3\mu^{2},\qquad\\{\Omega,X_{3}\\}=-2X_{1}\Omega,$ (57a) together with
the constraint $X_{3}^{2}=(3k_{2}^{2}H^{2}+2k_{1}(3\Omega-
X_{2})H-X_{1}^{2}+6\mu k_{2}H+3\mu^{2})\Omega-(k_{2}H+\mu)^{2}X_{2}.$ (57b)

In the reduction $k_{1}=1,\,k_{2}=0$, these integrals and relations just
reduce to those of Section 4.1.

##  9 Conclusions

This paper has continued the work of [12, 13]. Here we derive the consequences
of having one particular quadratic first integral $X_{1}$, which allows the
Calogero-Moser model as a particular case. This led to the Liouville
integrable triple (7), in terms of three arbitrary functions $A_{1},A_{2}$ and
$B$. The functions $A_{i}$ correspond to a 3D generalisation of one of the
standard 2D separable systems (in parabolic coordinates). The function $B$
allows for the connection to the Calogero-Moser system, corresponding to a
particular choice of $B$. This gives a coupling of the Calogero-Moser system
with a potential depending on 2 arbitrary functions, which includes (for
special choices of $A_{i}$) the Kepler potential, a resonant harmonic
oscillator, the (KdV related) Hénon-Heiles potential and many more. All of
these are at least Liouville integrable, but, in many cases, superintegrable.

This whole class of system was extended to the “conformally flat” case,
including the Kepler-Calogero-Moser system with a spherically symmetric
kinetic energy. However, the most interesting results came through symmetry
reduction, which mostly destroyed the connection with the Calogero-Moser
system. For example we presented a Kepler problem (33a) and a generalised
Hénon-Heiles system (34a) on a Darboux-Koenigs $D_{2}$ background. Other
systems reduced to a $D_{1},\,D_{3}$ or $D_{4}$ backgrounds.

Two and three dimensional systems have been studied a lot, so are quite well
understood. Some particular cases are known for general $n$, but these are
rare, mainly associated with Lax pairs (such as the rational Calogero-Moser
system and its generalisations). However, it would be interesting to
investigate how the construction of this paper could be generalised to 4 and
higher dimensions. In particular, the conformally flat case should lead to
interesting new features. As emphasised in [9, 11, 12], the conformal algebra
plays an important role in building both Liouville and superintegrable
systems. It is particularly important in the non-constant curvature case,
where Killing tensors (the leading order coefficients in higher order first
integrals) cannot always be built from Killing vectors. Symmetry reductions
(as in Section 7) should be particularly interesting, allowing us to reduce
from 4 to 3 to 2 dimensions in a variety of ways.

Of course, the hope is that those systems which exist for all $n$, will have a
Lax pair. However, constructing this Lax pair could be a difficult task.

Most superintegrable systems in classical mechanics have quantum counterparts,
although some problems can occur with higher order integrals [5, 9]. Most of
the emphasis is on deriving the spectrum and solving for eigenfunctions
through separation of variables [2, 18, 14]. However, as emphasised in [8,
10], superintegrability can be directly used for building eigenfunctions. For
many of the systems discussed in this paper, the analysis of the quantum
version is an open problem.

### Acknowledgements

This work was supported by the National Natural Science Foundation of China
(grant no. 11871396).

##  10 Appendix

###  10.1 Notation and Definitions

Here we collect some useful definitions and formulae, used throughout the
paper.

We give a detailed analysis of the conformal algebra in 3D in [12], but here
just emphasise a few important elements. For our purposes it is more
convenient to write all formulae in terms of Hamiltonian functions, rather
than as metric tensors and Killing vectors.

The symmetry algebra (a subalgebra of the conformal algebra) of the Kinetic
energy $H^{0}=\frac{1}{2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})$ just consists of
translations and rotations, represented by

$p_{1},\,p_{2},\,p_{3},\,J_{1}=q_{2}p_{3}-q_{3}p_{2},\,J_{2}=q_{3}p_{1}-q_{1}p_{3},\,J_{3}=q_{1}p_{2}-q_{2}p_{1}.$
(58a) An important conformal element is $S=q_{1}p_{1}+q_{2}p_{2}+q_{3}p_{3},$
(58b) satisfying $\\{S,H^{0}\\}=2H^{0}$.

The Casimir of the rotation algebra is

$J^{2}=J_{1}^{2}+J_{2}^{2}+J_{3}^{2}=2(q_{1}^{2}+q_{2}^{2}+q_{3}^{2})H^{0}-S^{2}.$
(58c)

The second representation, in terms of $H^{0}$ and $S$, is important when
adding potentials, $H=H^{0}+U(q_{1},q_{2},q_{3})$.

Any first integral of $H^{0}$ can be built from the “Killing vectors” (58a).
It is enough to consider homogeneous polynomials in the elements of (58a),
representing higher order Killing tensors. This is only true for flat and
constant curvature metrics. The second representation of $J^{2}$ in (58c) is
in terms of conformal Killing vectors and tensors and can be used in the
context of conformally flat metrics (see [9, 11]).

Integrals for $H=H^{0}+U(q_{1},q_{2},q_{3})$ have these “Killing tensors” as
leading order parts. A particularly important quadratic “Killing tensor” in
this paper is

$X_{1}^{0}=(p_{1}-p_{2})J_{3}+(p_{2}-p_{3})J_{1}+(p_{3}-p_{1})J_{2}=S(p_{1}+p_{2}+p_{3})-2(q_{1}+q_{2}+q_{3})H^{0}.$
(58d)

Again, this second representation of $X_{1}^{0}$ is important when extending
to $H=H^{0}+U(q_{1},q_{2},q_{3})$ and also when considering conformally flat
metrics.

######  Remark 10.1 (Coordinates $(Q_{i},P_{i})$)

Most of the calculations in this paper are performed within the
$(Q_{i},P_{i})$ coordinates. We use the same symbols
$J_{1}=Q_{2}P_{3}-Q_{3}P_{2}$, etc for the rotations in these coordinates, but
be warned that they are not invariant under the canonical transformation with
(3). We also use $S$ to mean $\sum_{i}Q_{i}P_{i}$. In this case the form of
$S$ is invariant under this transformation, as shown in Section 2.1.

###  10.2 Poisson Relations for Section 6.3.2

In Section 6.3.2 we considered the Poisson algebra associated with the
Hamiltonian (25a). We found that we could replace $X_{1}$, using
$X_{1}=-X_{5}-X_{6}$, and that the 10 dimensional Poisson algebra consisted of
6 quadratic functions $H,X_{2},\dots,X_{6}$ and 4 cubic ones
$X_{7},\dots,X_{10}$. The Poisson relations of the quadratic elements are
simple, so given in Section 6.3.2. Here we consider the remaining 24 Poisson
relations between quadratic and cubic functions, as well as the 6 Poisson
relations between cubic ones. This task is simplified by utilising a discrete
symmetry of the Hamiltonian, which is invariant under the involution
$\iota_{23}$ (which switches $Q_{2}$ and $Q_{3}$), if we extend this to act on
2 of the parameters: $k_{2}\leftrightarrow k_{4}$. This enabled us to reflect
this symmetry in our choice of functions, which satisfy:

$(H,X_{2},X_{3},X_{4},X_{5},X_{6},X_{7},X_{8},X_{9},X_{10},k_{2},k_{4})\mapsto(H,X_{2},X_{4},X_{3},X_{6},X_{5},-X_{7},-X_{8},X_{10},X_{9},k_{4},k_{2}).$

As a result, if we have the formula for $\\{X_{3},X_{9}\\}$, we can deduce the
formula of $\\{X_{4},X_{10}\\}$ by applying $\iota_{23}$. Introducing the
notation $P_{ij}=\\{X_{i},X_{j}\\}$, this means that $P_{39}$ and $P_{410}$
are paired. On the other hand, since $P_{910}\mapsto-P_{910}$, we obtain
nothing new. This is depicted in the list below:

$\displaystyle(P_{23},P_{24}),\quad(P_{25},P_{26}),\quad P_{27},\quad
P_{28},\quad(P_{29},P_{210}),\quad
P_{34},\quad(P_{35},P_{46}),\quad(P_{36},P_{45}),$
$\displaystyle(P_{37},P_{47}),\quad(P_{38},P_{48}),\quad(P_{39},P_{410}),\quad(P_{310},P_{49}),\quad
P_{56}\quad(P_{57},P_{67}),\quad(P_{58},P_{68}),$
$\displaystyle(P_{59},P_{610}),\quad(P_{510},P_{69}),\quad
P_{78},\quad(P_{79},P_{710}),\quad(P_{89},P_{810}),\quad P_{910}.$

The additional, independent entries in the Poisson matrix are

$\displaystyle P_{27}=2(X_{4}-X_{3})X_{2}+2(k_{2}-k_{4})(X_{3}+X_{4}),\quad
P_{28}=4(X_{6}-X_{5})X_{2}-4(k_{2}-k_{4})(X_{5}+X_{6}),$ $\displaystyle
P_{29}=4(X_{3}X_{6}-X_{4}X_{5}),\;\;P_{37}=-2X_{3}X_{4}+4\omega^{2}(X_{2}-k_{2}-k_{4}),\;\;P_{38}=4k_{3}(X_{2}-k_{2}-k_{4})H-4X_{4}X_{5},$
$\displaystyle P_{39}=4k_{3}HX_{3}-8\omega^{2}X_{5},\quad P_{310}=0,\quad
P_{57}=2k_{3}(X_{2}-k_{2}-k_{4})H-2X_{3}X_{6},$ $\displaystyle
P_{58}=4k_{1}(k_{2}+k_{4}-X_{2})H+3(X_{2}-k_{2}-k_{4})X_{3}+2(X_{2}-2k_{2}-k_{4})X_{4}-4X_{5}X_{6},$
$\displaystyle
P_{59}=-4(k_{1}X_{3}+k_{3}X_{5})H+(3X_{3}+2X_{4})X_{3}-4k_{2}\omega^{2},\quad
P_{510}=X_{3}X_{4}-2\omega^{2}(X_{2}-k_{2}-k_{4}),$ $\displaystyle
P_{78}=4(X_{6}-X_{5})X_{7}+2(X_{3}-X_{4})X_{8}+4k_{2}X_{10}+4k_{4}X_{9},\quad
P_{79}=4k_{3}HX_{7}+2X_{4}X_{9}-4\omega^{2}X_{8},$ $\displaystyle
P_{89}=-4(2k_{1}X_{7}+k_{3}X_{8})H+2(3X_{3}+2X_{4})X_{7}+4X_{6}X_{9},\quad
P_{910}=4\omega^{2}X_{7}.$

The ten integrals of the Poisson algebra are of rank 5 and satisfy the
following five constraints

$\displaystyle 2k_{3}HX_{7}-2\omega^{2}X_{8}-X_{10}X_{3}+X_{4}X_{9}=0,$
$\displaystyle(2k_{1}H-X_{3}-X_{4})X_{7}+k_{3}HX_{8}+X_{10}X_{5}-X_{6}X_{9}=0,$
$\displaystyle
2(X_{5}-X_{6})X_{7}+(X_{4}-X_{3})X_{8}+(X_{2}-k_{2}-3k_{4})X_{9}+(X_{2}-3k_{2}-k_{4})X_{10}=0,$
$\displaystyle(X_{2}-k_{2}-k_{4})(X_{10}-X_{9})+(X_{3}+X_{4})X_{8}-2(X_{5}+X_{6})X_{7}-2k_{4}X_{9}+2k_{2}X_{10}=0,$
$\displaystyle
X_{7}^{2}-X_{2}X_{3}X_{4}+k_{4}X_{3}^{2}+k_{2}X_{4}^{2}+(k_{2}+k_{4})X_{3}X_{4}+\omega^{2}\left((X_{2}-k_{2}-k_{4})^{2}-4k_{2}k_{4}\right)=0.$

###  10.3 Poisson Relations for Section 3.4.1

The Hamiltonian (25a) reduces to its flat counterpart (12a), with condition
(13d), when $k_{1}=1,\,k_{3}=0$, along with the change of notation
$(b_{1},b_{2})=\left(\frac{1}{2}k_{2},\frac{1}{2}k_{4}\right)$. Therefore, the
Poisson algebra for the flat case is obtained from that of (25a), by this
simple parametric reduction.

The definitions of $X_{i}$ and the Poisson relations, given in Section 6.3.2
are unchanged, but some of the $P_{ij}$, given above, are simplified.
Furthermore, the above five constraints can be replaced by the (mainly)
simpler formulae:

$\displaystyle
X_{4}X_{9}-X_{10}X_{3}-2\omega^{2}X_{8}=0,\quad(2H-X_{3}-X_{4})X_{7}+X_{10}X_{5}-X_{6}X_{9}=0,$
$\displaystyle
X_{7}^{2}-X_{2}X_{3}X_{4}+k_{4}X_{3}^{2}+k_{2}X_{4}^{2}+(k_{2}+k_{4})X_{3}X_{4}+\omega^{2}\left((X_{2}-k_{2}-k_{4})^{2}-4k_{2}k_{4}\right)=0,$
$\displaystyle
X_{9}^{2}+4\omega^{2}X_{5}^{2}+(X_{3}+X_{4}-2H)(X_{3}^{2}-4\omega^{2}k_{2})=0,$
$\displaystyle
X_{10}^{2}+4\omega^{2}X_{6}^{2}+(X_{3}+X_{4}-2H)(X_{4}^{2}-4\omega^{2}k_{4})=0.$

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