# A note on the integrability of exceptional potentials via polynomial bi-
homogeneous potentials

Primitivo B. Acosta–Humánez Instituto de Matemática, Facultad de Ciencias,
Universidad Autónoma de Santo Domingo, Dominican Republic. (pacosta-
humanez@uasd.edu.do), (ORCID ID: 0000-0002-5627-4188). ∗Corresponding Author.
Martha Álvarez–Ramírez Departamento de Matemáticas, UAM–Iztapalapa, México,
City, México. (mar@xanum.uam.mx), (ORCID ID: 0000-0001-9187-1757)
Teresinha J. Stuchi Instituto de Física, Universidade Federale de Rio de
Janeiro, Rio de Janeiro, Brazil (tstuchi@if.ufjr.br), (ORCID ID:
0000-0003-0700-6205)

###### Abstract

This paper is concerned with the polynomial integrability of the two-
dimensional Hamiltonian systems associated to complex homogeneous polynomial
potentials of degree $k$ of type
$V_{k,l}=\alpha(q_{2}-iq_{1})^{l}(q_{2}+iq_{1})^{k-l}$ with
$\alpha\in\mathbb{C}$ and $l=0,1,\dots,k$, called exceptional potentials.
Hietarinta [9] proved that the potentials with $l=0,1,k-1,k$ and $l=k/2$ for
$k$ even are polynomial integrable. We present an elementary proof of this
fact in the context of the polynomial bi-homogeneous potentials, as was
introduced by Combot et al. [8]. In addition, we take advantage of the fact
that we can exchange the exponents to derive an additional first integral for
$V_{7,5}$, unknown so far. The paper concludes with a Galoisian analysis for
$l=k/2$.

###### keywords:

Hamiltonian system with two degrees of freedom, Homogeneous potentials,
Integrability.

Resumen

Este artículo trata de la integrabilidad polinomial de sistemas hamiltonianos
potenciales polinomiales homogeneos complejos de grado $k$ del tipo
$V_{k,l}=\alpha(q_{2}-iq_{1})^{l}(q_{2}+iq_{1})^{k-l}$ con
$\alpha\in\mathbb{C}$ y $l=0,1,\dots,k$, denominados potenciales
excepcionales. Hietarinta [9] probó que los potenciales con $l=0,1,k-1,k$ y
$l=k/2$ para $k$ par son integrables polinomialmente. Presentamos una prueba
elemental de este hecho en el contexto de los potenciales bi-homogéneos
polinomiales, tal como fueron introducidos por Combot et al. [8].
Adicionalmente, tomamos ventaja del hecho que podemos intercambiar los
exponentes para obtener una integral primera adicional para $V_{7,5}$,
desconocido hasta donde sabemos. El artículo finaliza con un análisis
Galoisiano para $l=k/2$.

Palabras claves: Sistemas hamiltonianos de dos grados de libertad, potenciales
homogéneos, integrabilidad.

##  1 Introduction

The existence of first integrals in Hamiltonian systems is of utmost
importance, since it allows us to lower the dimension of such systems. The
Liouville theorem states that a Hamiltonian system with $n$ degrees of freedom
is completely integrable if there exist $n$ independent first integrals in
involution. Thus, the problem can be integrated by quadratures, see [1]. In
other words, completely integrable Hamiltonians are systems for which, in a
way or another, we can obtain a full description of their solutions

The study of the integrability of Hamiltonian systems with homogeneous
potentials within the framework of Ziglin’s theory [20, 21] was started by
Yoshida [18, 19]. Since then, a great deal of research has been done, mainly
based on differential Galois theory of ordinary linear differential equations,
where the Morales-Ramis theorem plays a key role giving necessary conditions
for the integrability in the Liouville sense of a Hamiltonian system having
meromorphic first integrals. See [13, 12, 17] and references therein.

We assume that $\mathbb{C}^{2n}$ is the symplectic linear space with
$\mathbf{q}=(q_{1},\dots,q_{n})$ and $\mathbf{p}=(p_{1},\dots,p_{n})$ being
the position and momentum vectors, respectively. Suppose we have a Hamiltonian
system with Hamiltonian function

$H=\frac{1}{2}\sum_{i=1}^{n}p_{i}^{2}+V(\mathbf{q}),$ (1)

where the potential energy $V=V(\mathbf{q})\in\mathbb{C}[\mathbf{q}]$ is a
homogeneous polynomial of degree $k$, that is,
$V(\lambda\mathbf{q})=\lambda^{k}V(\mathbf{q})$ for all nonzero
$\lambda\in\mathbb{C}$ and $k\in\mathbb{Z}$. The motion equations of the above
Hamiltonian can be written as:

$\frac{d}{dt}\mathbf{q}=\mathbf{p},\qquad\frac{d}{dt}\mathbf{p}=-V^{\prime}(\mathbf{q}),$
(2)

where $V^{\prime}(\mathbf{q})$ indicates the gradient of $V(\mathbf{q})$. The
concept of Darboux points play a central role in the integrability analysis of
a homogeneous potential. A non-zero point $\mathbf{d}\in\mathbb{C}^{n}$ is
called a _Darboux point_ of (2) if it is a solution of
$V^{\prime}(\mathbf{q})=\lambda\mathbf{d}$, where
$\lambda\in\mathbb{C}\setminus\\{0\\}$. It is also said that a Darboux point
$\mathbf{d}$ is a proper Darboux point if and only if
$V^{\prime}(\mathbf{d})\neq\mathbf{0}$. Otherwise, it is called an improper
Darboux point. In other words, a Darboux point can be seen as a point in the
complex projective space $\mathbb{C}\mathbb{P}^{n-1}$ with
$[d_{1}:\cdots:d_{n}]$ as homogeneous coordinates. For more information on
this subject the reader is referred to [12].

From now on, we shall constrain ourselves to the Hamiltonian systems with two
degrees of freedom. At this stage, we consider the group of complex matrices
$A_{2\times 2}\in{\rm PO}(2,\mathbb{C})\cong\mathbb{C}\mathbb{P}^{1}$. Here
${\rm PO}(2,\mathbb{C})$ denotes the group of $2\times 2$ complex matrices $A$
such that $AA^{T}=\alpha I_{2}$, where $\alpha\in\mathbb{C}\setminus\\{0\\}$
and $I_{2}$ is the $2$-dimensional identity matrix. The potentials
$V_{1}({\mathbf{q}})$ and $V_{2}({\mathbf{q}})$ are said to be equivalents if
there exists a matrix $A\in{\rm PO}(2,\mathbb{C})$ such that
$V_{1}({\mathbf{q}})=V_{2}(A{\mathbf{q}})$. This concept was inspired by a
Hietarinta’s result proved in [10], namely that if $V_{1}({\mathbf{q}})$ is
integrable, then $V_{2}(A{\mathbf{q}})$ is also integrable for any $A\in{\rm
PO}(2,\mathbb{C})$.

A homogeneous polynomial $V\in\mathbb{C}[q_{1},q_{2}]$ of degree $k$ is a
polynomial such that
$V(q_{1},q_{2})=q^{k}_{1}V(1,q_{2}/q_{1})=q_{2}^{k}V(q_{1}/q_{2},1)$. Let be
$v(x):=V(1,x)$ and $w(\xi):=V(\xi,1)$. Since ${\rm
PO}(2,\mathbb{C})=\mathbb{C}\mathbb{P}^{1}$, we use $[q_{1}:q_{2}]$ as
homogeneous coordinates on $\mathbb{C}\mathbb{P}^{1}$. Moreover, points
$[1:q_{2}]\in\mathbb{C}\mathbb{P}^{1}$ will be identified with the affine part
of $\mathbb{C}\mathbb{P}^{1}$ when it is parameterized by coordinates
$x=q_{2}/q_{1}$, $q_{1}\neq 0$. The points in a neighborhood of infinity are
parameterized by the other coordinate $\xi=q_{1}/q_{2}$. In [12], the authors
choose amid all equivalent points a representative $V$, whose polynomial $w$
(or $v$) has one zero at some point of
$\mathbb{C}\mathbb{P}^{1}\setminus\\{[1:+i],[1:-i]\\}$. This always happens,
except for the named exceptional potentials given by

$V_{k,l}(q_{1},q_{2})=\alpha(q_{2}-iq_{1})^{l}(q_{2}+iq_{1})^{k-l},\;\mbox{where}\;l=0,\dots,k\;\mbox{and}\;\alpha\in\mathbb{C}\setminus\\{0\\}.$

Indeed, it follows that these potentials lack Darboux points, see [12]. So,
Morales-Ramis theory concerning homogeneous potentials cannot be applied to
investigate their integrability.

The goal of this paper is to give an step forward in the issue of existence of
polynomial first integrals for two degrees of freedom complex Hamiltonian
systems with Hamiltonian function of the form

$H=\frac{1}{2}(p_{1}^{2}+p_{2}^{2})+V_{k,l}(q_{1},q_{2}),$ (3)

with exceptional homogeneous polynomial potentials of degree $k$ given by

$V_{k,l}(q_{1},q_{2})=\alpha(q_{2}-iq_{1})^{l}(q_{2}+iq_{1})^{k-l},\hskip
34.14322ptl=0,1,\dots,k,$ (4)

where $k$ is a positive integer. For more details, see [12].

As the Hamiltonian (3) is time independent and the motion takes place on the
submanifold $\mathbb{C}^{2n}$ defined by a constant value $H$, which means
that $H$ itself is always a first integral, and since the Hamiltonian (3) has
two degree of freedom, our system will be completely (or Liouville) integrable
if we can find a second first integral $F$ functionally independent of the
Hamiltonian function $H,$ satisfying that $\\{H,F\\}=0$; that is, if $F$ and
$H$ commute under the Poisson bracket. For more information the reader is
referred to [1].

In [9], Hietarinta proved that the Hamiltonian systems with the exceptional
potentials $V_{k,0}$, $V_{k,1}$, $V_{k,k-1}$, $V_{k,k}$ and $V_{k,k/2}$ for
even $k$ are polynomial integrable, that is, they have an additional
polynomial first integral functionally independent of the Hamiltonian. As a
consequence, all exceptional potentials for $k\leq 4$ are completely
integrable with a polynomial integral besides the classical first integral.
Additionally, Llibre and Valls in [11] proved that Hamiltonian systems having
potential $V_{k,l}$ with even $k\geq 6$ and $l=2,\dots,k/2-1,k/2+1,\dots,k-2$,
do not admit an additional polynomial first integral. They performed a non
symplectic change of variable to reduce the issue of existence of analytic
first integrals in the Hamiltonian system to an equivalent weight-homogeneous
polynomial differential system. Also, the authors in [11] claimed that nothing
is known about the integrability in the case when $k$ is odd, which still
remains as an open problem. Nevertheless, Nakagawa et al. [15] used the so-
called direct method to obtain an additional polynomial first integral of
degree 4 in the momenta for the exceptional potential of degree seven with
$l=2$, expressed as $V_{7,2}=(q_{2}-iq_{1})^{2}(q_{2}+iq_{1})^{5}$.
Furthermore, Nakagawa and Yoshida [16] claimed that by using the direct method
they have checked that for $5\leq k\leq 20$, $k\neq 7$, the Hamiltonian system
with two degree of freedom and potential $V_{k,l}$ and
$l\not\in\\{0,1,k/2\\}$, does not admit a further first polynomial integral
with degree smaller than 9 in the momenta.

##  2 Exceptional potentials viewed as polynomial bi-homogeneous potentials

In [8], Combot et al. used a theorem by Moulin Ollagnier about integrability
of the homogeneous Lotka-Volterra system (see [14]), that allowed them to make
a complete classification of integrable cases of Hamiltonian systems with two
degrees of freedom endowed with polynomial bi-homogeneous potentials. As a
result, the necessary conditions for the integrability are transformed in a
way that leads to investigate the integrability of a family of bi-homogeneous
potentials which depend on two integer parameters. This result inspired us to
see an exceptional potential as a polynomial bi-homogeneous potential, in
order to recover in a straight way the polynomial integrability of Hamiltonian
system with potential $V_{k,l}$ with $l=0,1,k-1,k$ and $l=k/2$ for $k$ even.

Consider the symplectic change of coordinates $x_{1}=q_{1}+iq_{2}$,
$y_{1}=\frac{1}{2}(p_{1}-ip_{2})$, $x_{2}=q_{1}-iq_{2}$,
$y_{2}=\frac{1}{2}(p_{1}+ip_{2})$. Through some direct calculations, we obtain
that the Hamiltonian is transformed into

$H=2y_{1}y_{2}+\beta\;x_{1}^{l}x_{2}^{k-l},$ (5)

where $\beta=(-1)^{l}i^{k}\alpha$, and the potential is given by

$V_{k,l}=\beta\;x_{1}^{l}x_{2}^{k-l},$ (6)

which is usually called polynomial bi-homogeneous. An application of the
Combot et al. theorems formulated below, will allow us to deduce our result.

######  Theorem 1 (Combot et al. in [8]).

The Hamiltonian system
$H(x_{1},x_{2},y_{1},y_{2})=2y_{1}y_{2}+x_{1}^{k_{1}}x_{2}^{k_{2}}$ with
$(k_{1},k_{2})\in\mathbb{Z}^{2}$ is rationally integrable if and only if
either $(k_{1},k_{2})$ or $(k_{2},k_{1})$ belongs to the following list

$(0,k),\qquad(1,k),\qquad(k,k),\qquad(k,-2-k),\qquad(2,5),\qquad
k\in\mathbb{Z}.$

######  Theorem 2 (Combot et al. [8]).

The Hamiltonian system given by
$H(x_{1},x_{2},y_{1},y_{2})=2y_{1}y_{2}+x_{1}^{k_{1}}x_{2}^{k_{2}}$ with
$(k_{1},k_{2})\in\mathbb{Q}^{2}$ is algebraically integrable if and only if

  1. 1.

$(k_{1},k_{2})=(0,k)$, the additional first integral being $J=y_{1}$.

  2. 2.

$(k_{1},k_{2})=(k,k)$, $J=y_{1}x_{1}-y_{2}x_{2}$.

  3. 3.

$(k_{1},k_{2})=(-\frac{1}{2},k)$,
$J=y_{1}(y_{1}x_{1}-y_{2}x_{2})-\frac{1}{2}x_{1}^{-1/2}x_{2}^{k+1}$.

  4. 4.

$(k_{1},k_{2})=(1,k)$, $J=y_{1}^{2}+\frac{1}{k+1}x_{2}^{k+1}$.

  5. 5.

$(k_{1},k_{2})=(-k-2,k)$,
$J=(y_{1}x_{1}-y_{2}x_{2})^{2}-2x_{1}^{-k-1}x_{2}^{k+1}$.

  6. 6.

$(k_{1},k_{2})=(2,5)$,
$J=6y_{1}^{3}(y_{1}x_{1}-y_{2}x_{2})+x_{2}^{6}(4y_{1}^{2}x_{1}^{2}-8y_{1}y_{2}x_{1}x_{2}+y_{2}^{2}x_{2}^{2})-x_{1}^{3}x_{2}^{12}$.

  7. 7.

$(k_{1},k_{2})=(2,-7/4)$,

$J=y_{1}^{6}+8x_{2}^{-3/4}y_{1}^{3}(4y_{2}x_{2}-y_{1}x_{1})+8x_{2}^{-3/2}(5y_{1}^{2}x_{1}^{2}-4y_{1}y_{2}x_{1}x_{2}+8y_{2}^{2}x_{2}^{2})-48x_{1}^{3}x_{2}^{-9/4},$

  8. 8.

$(k_{1},k_{2})=(2,-10/7)$,

$J=4y_{1}^{6}+14x_{2}^{-3/7}y_{1}^{3}(7y_{2}x_{2}-4y_{1}x_{1})+343x_{2}^{-6/7}(y_{1}x_{1}-7y_{2}x_{2})(y_{1}x_{1}-y_{2}x_{2})-2058x_{1}^{3}x_{2}^{-9/7},$

  9. 9.

$(k_{1},k_{2})=(3,-9/5)$,

$J=4y_{1}^{4}-10x_{2}^{-4/5}(3y_{1}^{2}x_{1}^{2}-30y_{1}y_{2}x_{1}x_{2}+25y_{2}^{2}x_{2}^{2})+225x_{1}^{4}x_{2}^{-8/5},$

  10. 10.

$(k_{1},k_{2})=(-2/3,-7/3)$,

$J=8y_{1}(y_{1}x_{1}-y_{2}x_{2})^{3}-x_{1}^{-2/3}x_{2}^{-4/3}(13y_{1}^{2}x_{1}^{2}-44y_{1}y_{2}x_{1}x_{2}+4y_{2}^{2}x_{2}^{2})+16x_{1}^{-1/3}x_{2}^{-8/3},$

  11. 11.

$(k_{1},k_{2})=(-2/3,-10/3)$,

$J=4y_{1}^{2}(y_{1}x_{1}-y_{2}x_{2})^{4}+2x_{1}^{-2/3}x_{2}^{-7/3}y_{1}(-4y_{1}^{3}x_{1}^{3}+13y_{1}^{2}y_{2}x_{1}^{2}x_{2}-20y_{1}y_{2}^{2}x_{1}x_{2}^{2}+2y_{2}^{3}x_{2}^{3}),$

  12. 12.

$(k_{1},k_{2})=(-2/3,-5/6)$,

$J=2x_{1}^{-2/3}(y_{1}x_{1}-y_{2}x_{2})^{2}[x_{2}^{1/6}-2y_{1}x_{1}^{2/3}(y_{1}x_{1}-y_{2}x_{2})]+x_{1}^{-1/3}x_{2}^{1/3},$

  13. 13.

$(k_{1},k_{2})=(-3/4,-9/4)$,

$J=4y_{1}(y_{1}x_{1}-y_{2}x_{2})^{3}-2x_{1}^{-3/4}x_{2}^{-5/4}(3y_{1}^{2}x_{1}^{2}-12y_{1}y_{2}x_{1}x_{2}+y_{2}^{2}x_{2}^{2})+9x_{1}^{-1/2}x_{2}^{-5/2}.$

At this stage the reader should be warned that we are interested in studying
polynomial potentials, as a consequence we will only consider the cases of
items from 1 to 6.

Since the potentials (4) and (6) are equivalent, we are thus led to the
following consequence of Theorems 1 and 2.

######  Proposition 1.

The Hamiltonian system with Hamiltonian function given by (5) is polynomially
integrable if and only if potentials and additional first integrals $J$ are
given by

  1. 1.

$V_{k,0}=\beta\>x_{2}^{k}$, $\beta=i^{k}\alpha$, $J=y_{1}$.

  2. 2.

$V_{k,k}=\beta\>x_{1}^{k}$, $\beta=(-i)^{k}\alpha$, $J=y_{2}$.

  3. 3.

$V_{k,k/2}=\beta x_{1}^{k/2}x_{2}^{k/2}$, with even $k$, $\beta=\alpha$,
$J=y_{1}x_{1}-y_{2}x_{2}$.

  4. 4.

$V_{k,1}=\beta\>x_{1}x_{2}^{k-1}$, $\beta=-i^{k}\alpha$,
$J=y_{1}^{2}-\dfrac{1}{k}i^{k}\alpha x_{2}^{k}$.

  5. 5.

$V_{k,k-1}=\beta\>x_{1}^{k-1}x_{2}$, $\beta=(-1)^{k-1}i^{k}\alpha$,
$J=y_{2}^{2}+\dfrac{1}{k}(-1)^{k+1}i^{k}\alpha x_{1}^{k}$.

  6. 6.

$V_{7,2}=\beta x_{1}^{2}x_{2}^{5}$, $\beta=-i\alpha$, $\alpha=i$,
$J=16y_{1}^{3}(y_{1}x_{1}-y_{2}x_{2})+x_{2}^{6}(4y_{1}^{2}x_{1}^{2}-8y_{1}y_{2}x_{1}x_{2}+y_{2}^{2}x_{2}^{2})-x_{1}^{3}x_{2}^{12}$.

  7. 7.

$V_{7,5}=\beta x_{1}^{5}x_{2}^{2}$, $\beta=i\alpha$, $\alpha=-i$,
$J=16y_{2}^{3}(y_{2}x_{2}-y_{1}x_{1})+x_{1}^{6}(4y_{2}^{2}x_{2}^{2}-8y_{2}y_{1}x_{1}x_{2}+y_{1}^{2}x_{1}^{2})-x_{2}^{3}x_{1}^{12}$.

###### Proof.

The function $J$ is a first integral of the Hamiltonian system with
Hamiltonian function (5) given by

$H(x_{1},x_{2},y_{1},y_{2})=2y_{1}y_{2}+V_{k,l}(x_{1},x_{2}),$

if the Poisson bracket

$\\{H,J\\}=\frac{\partial H}{\partial y_{1}}\frac{\partial J}{\partial
x_{1}}-\frac{\partial H}{\partial x_{1}}\frac{\partial J}{\partial
y_{1}}+\frac{\partial H}{\partial y_{2}}\frac{\partial J}{\partial
x_{2}}-\frac{\partial H}{\partial x_{2}}\frac{\partial J}{\partial y_{2}}$

is zero. Thence, the proof of the statements in the theorem relies on a direct
construction of $\\{H,J\\}$ for each one-item in the list given below.

  1. 1.

$V_{k,0}=i^{k}\alpha\>x_{2}^{k}$, $J=y_{1}$

$\\{H,J\\}=2y_{2}\cdot 0+2y_{1}\cdot 0+0\cdot 1-ki^{k}\alpha x_{2}{k-1}\cdot
0=0.$

  2. 2.

$V_{k,k}=(-i)^{k}\alpha\>x_{1}^{k}$, $J=y_{2}$

$\\{H,J\\}=2y_{2}\cdot 0+(-1)^{k+1}ki^{k}\alpha x_{1}{k-1}\cdot 0+2y_{1}\cdot
0+0\cdot 1=0.$

  3. 3.

$V_{k,k/2}=\alpha x_{1}^{k/2}x_{2}^{k/2}$, $J=y_{1}x_{1}-y_{2}x_{2}$

$\\{H,J\\}=2y_{2}y_{1}-\frac{1}{2}k\alpha
x_{1}^{\frac{k}{2}-1}x_{2}^{k/2}\cdot x_{1}-2y_{1}y_{2}-\frac{1}{2}k\alpha
x_{1}^{k/2}x_{2}^{\frac{k}{2}-1}\cdot(-x_{2})=0.$

  4. 4.

$V_{k,1}=-i^{k}\alpha\>x_{1}x_{2}^{k-1}$, $J=y_{1}^{2}-\dfrac{1}{k}i^{k}\alpha
x_{2}^{k}$

$\displaystyle\\{H,J\\}$ $\displaystyle=$ $\displaystyle 2y_{2}\cdot
0+i^{k}\alpha x_{2}^{k-1}\cdot(2y_{1})+2y_{1}\cdot(-\alpha
i^{k}x_{2}^{k-1})+i^{k}(k-1)\alpha x_{1}x_{2}^{k-2}\cdot 0$ $\displaystyle=$
$\displaystyle 2i^{k}\alpha x_{2}^{k-1}y_{1}-2\alpha i^{k}x_{2}^{k-1}y_{1}=0$

  5. 5.

$V_{k,k-1}=(-1)^{k-1}i^{k}\alpha\>x_{1}^{k-1}x_{2}$,
$J=y_{2}^{2}+\dfrac{1}{k}(-1)^{k+1}i^{k}\alpha x_{1}^{k}$

$\displaystyle\\{H,J\\}$ $\displaystyle=$ $\displaystyle
2y_{2}\cdot(-1)^{k+1}i^{k}\alpha x_{1}^{k-1}+(-1)^{k}i^{k}(k-1)\alpha
x_{1}^{k-2}x_{2}\cdot 0+2y_{1}\cdot 0+(-1)^{k}i^{k}\alpha
x_{1}^{k-1}\cdot(2y_{2})$ $\displaystyle=$ $\displaystyle 0$

  6. 6.

$V_{7,2}=x_{1}^{2}x_{2}^{5}$,
$J=16y_{1}^{3}(y_{1}x_{1}-y_{2}x_{2})+x_{2}^{6}(4y_{1}^{2}x_{1}^{2}-8y_{1}y_{2}x_{1}x_{2}+y_{2}^{2}x_{2}^{2})-x_{1}^{3}x_{2}^{12}$

$\displaystyle\\{H,J\\}$ $\displaystyle=$
$\displaystyle-6x_{1}^{2}x_{2}^{12}y_{2}+16x_{1}x_{2}^{6}y_{1}^{2}y_{2}+32y_{1}^{4}y_{2}-16x_{2}^{7}y_{1}y_{2}^{2}$
$\displaystyle\qquad-\;16x_{1}^{3}x_{2}^{11}y_{1}-128x_{1}^{2}x_{2}^{5}y_{1}^{3}+16x_{1}^{2}x_{2}^{12}y_{2}+96x_{1}x_{2}^{6}y_{1}^{2}y_{2}$
$\displaystyle\qquad-\;24x_{1}^{3}x_{2}^{11}y_{1}+48x_{1}^{2}x_{2}^{5}y_{1}^{3}-112x_{1}x_{2}^{6}y_{1}^{2}y_{2}-32y_{1}^{4}y_{2}+16x_{2}^{7}y_{1}y_{2}^{2}$
$\displaystyle\qquad+\;40x_{1}^{3}x_{1}^{11}y_{1}+80x_{1}^{2}x_{2}^{5}y_{1}^{3}-10x_{1}^{2}x_{2}^{12}y_{2}=0$

  7. 7.

$V_{7,5}=x_{1}^{5}x_{2}^{2}$,
$J=16y_{2}^{3}(y_{2}x_{2}-y_{1}x_{1})+x_{1}^{6}(4y_{2}^{2}x_{2}^{2}-8y_{2}y_{1}x_{1}x_{2}+y_{1}^{2}x_{1}^{2})-x_{2}^{3}x_{1}^{12}$

$\displaystyle\\{H,J\\}$ $\displaystyle=$ $\displaystyle
2y_{2}\cdot(-4)(3x_{1}^{11}x_{2}^{3}-2x_{1}^{7}y_{1}^{2}+14x_{1}^{6}x_{2}y_{1}y_{2}-6x_{1}^{5}x_{2}^{2}y_{2}^{2}+4y_{1}y_{2}^{3}$
$\displaystyle\qquad-\;5x_{1}^{4}x_{2}^{2}\cdot
2(x_{1}^{8}y_{1}-4x_{1}^{7}x_{2}y_{2}-8x_{1}y_{2}^{3})$
$\displaystyle\qquad+\;2y_{1}\cdot(-3x_{1}^{12}x_{2}^{2}-8x_{1}^{7}y_{1}y_{2}+8x_{1}^{6}x_{2}y_{2}^{2}+16y_{2}^{4})$
$\displaystyle\qquad-\;2x_{1}^{5}x_{2}\cdot
8(-x_{1}^{7}x_{2}y_{1}+x_{1}^{6}x_{2}^{2}y_{2}-6x_{1}y_{1}y_{2}^{2}+8x_{2}y_{2}^{3})=0$

∎

######  Remark 1.

We can observe that
$V_{7,2}(x_{1},x_{2},y_{1},y_{2})=V_{7,5}(x_{2},x_{1},y_{2},y_{1})$. That is,
there exists a permutation $\theta\in S_{4}$ such that
$V_{7,5}(x_{1},x_{2},y_{1},y_{2})=V_{7,2}(\theta(x_{1}),\theta(x_{2}),\theta(y_{1}),\theta(y_{2}))$,
where the permutation $\theta$ and permutation matrix $A_{\theta}$ are given
respectively by

$\theta=\begin{pmatrix}1&2&3&4\\\ 2&1&4&3\end{pmatrix}=(1,2)(3,4),\quad
A_{\theta}=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}$

and $x_{1}\mapsto 1$, $x_{2}\mapsto 2$, $y_{1}\mapsto 3$ and $y_{2}\mapsto 4$.
Now, setting the linear transformations
$T(q_{1},q_{2},p_{1},p_{2})=\frac{1}{2}(q_{1}+iq_{2},q_{1}-iq_{2},p_{1}-ip_{2},p_{1}+ip_{2})$
and $T_{\theta}$ with companion matrix $A_{\theta}$, we obtain
$T_{\theta}\circ T\circ
T_{\theta}^{-1}(q_{1},q_{2},p_{1},p_{2})=(q_{2},q_{1},p_{2},p_{1})$. Thus, we
can see that in coordinates $(q_{1},q_{2},p_{1},p_{2})$ it is satisfied that
$V_{7,5}(q_{1},q_{2},p_{1},p_{2})=V_{7,5}(q_{2},q_{1},p_{2},p_{2})$.

Finally, we summarize all the result of the Proposition 1 in terms of the
variables $(q_{1},q_{2},p_{1},p_{2})$ in Table 1.

Table 1: Exceptional potential $V_{k,l}$ and the additional first integral $J$ in terms of their natural variables $(q_{1},q_{2},p_{1},p_{2})$. Exceptional potential $V_{k,l}$ |  First integral $J$
---|---
$V_{k,0}=\alpha(q_{2}+iq_{1})^{k}$ | $\frac{1}{2}(p_{1}-ip_{2})$
$V_{k,k}=\alpha(q_{2}-iq_{1})^{k}$ | $\frac{1}{2}(p_{1}+ip_{2})$
$V_{k,k/2}=\alpha(q_{2}-iq_{1})^{k/2}(q_{2}+iq_{1})^{k/2}$ | $-\frac{i}{2}(p_{1}q_{2}-q_{1}p_{2})$
$V_{k,1}=\alpha\>(q_{2}-iq_{1})(q_{2}+iq_{1})^{k-1}$ | $\frac{1}{4}(p_{1}-ip_{2})^{2}-\dfrac{1}{k}i^{k}\alpha(q_{1}-iq_{2})^{k}$
$V_{k,k-1}=\alpha(q_{2}-iq_{1})^{k-1}(q_{2}+iq_{1})$ | $\frac{1}{4}(p_{1}+ip_{2})^{2}+\dfrac{1}{k}(-1)^{k+1}i^{k}\alpha(q_{1}+iq_{2})^{k}$
$V_{7,2}=(q_{1}-iq_{2})^{5}(q_{1}+iq_{2})^{2}$ | $-(p_{1}-ip_{2})^{3}(p_{1}+ip_{2})(q_{1}-iq_{2})+\frac{1}{4}(p_{1}+ip_{2})^{2}(q_{1}-iq_{2})^{8}$
|
$+(p_{1}-ip_{2})^{4}(q_{1}+iq_{2})-2(p_{1}-ip_{2})(p_{1}+ip_{2})(q_{1}-iq_{2})^{7}(q_{1}+iq_{2})$
|
$+(p_{1}-ip_{2})^{2}(q_{1}-iq_{2})^{6}(q_{1}+iq_{2})^{2}-(q_{1}-iq_{2})^{12}(q_{1}+iq_{2})^{3}$
$V_{7,5}=(q_{1}-iq_{2})^{2}(q_{1}+iq_{2})^{5}$ | $(p_{1}+ip_{2})^{4}(q_{1}-iq_{2})-(p_{1}-ip_{2})(p_{1}+ip_{2})^{3}(q_{1}+iq_{2})$
| $+(p_{1}+ip_{2})^{2}(q_{1}-iq_{2})^{2}(q_{1}+iq_{2})^{6}$
| $-2(p_{1}-ip_{2})(p_{1}+ip_{2})(q_{1}-iq_{2})(q_{1}+iq_{2})^{7}$
|
$+\frac{1}{4}(p_{1}-ip_{2})^{2}(q_{1}+iq_{2})^{8}-(q_{1}-iq_{2})^{3}(q_{1}+iq_{2})^{12}$

Let us conclude by saying that our results agree well with those of Hietarinta
[10] and Nakagawa et al. [15]. Although it is obvious that
$V_{7,5}(x_{1},x_{2},y_{1},y_{2})=V_{7,2}(x_{2},x_{1},y_{2},y_{1})$, it is not
evident that
$V_{7,5}(q_{1},q_{2},p_{1},p_{2})=V_{7,2}(q_{2},q_{1},p_{2},p_{1})$ due to we
need the conjugation of $T$ with $T_{\theta}$. Moreover, as far as the authors
are aware, this is the first time that an additional polynomial first integral
for $V_{7,5}(q_{1},q_{2},p_{1},p_{2})$, different from the Hamiltonian one, is
given explicitly.

##  3 Algebraic Analysis

We start this section summarizing for reader’s convenience the major facts on
algebraic aspects, invariant planes and differential Galois group of the first
variational equation. For more information on this subject the reader is
referred to [2, 3, 4, 5, 6, 7].

The starting point is the concept of differential field. A differential field
$K$ is a field endowed with a derivation $\partial$, such that for all $a,b\in
K$ it satisfied:

  1. 1.

$\partial(a+b)=\partial a+\partial b$

  2. 2.

$\partial(a\cdot b)=a\cdot\partial b+\partial a\cdot b$

  3. 3.

$\partial\left(\dfrac{a}{b}\right)=\dfrac{1}{b^{2}}(a\cdot\partial b-\partial
a\cdot b)$.

The field of constants of $K$, denoted by $C_{K}$, is given by

$C_{K}=\\{c\in K:\,\,\partial(c)=0\\}.$

Consider the system of linear differential equations over $K$ given by

$\frac{d}{dx}\begin{pmatrix}y_{1}\\\ y_{2}\\\ \vdots\\\
y_{n}\end{pmatrix}=\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\\
a_{21}&a_{22}&\ldots&a_{2n}\\\ \vdots&&&\\\ a_{n1}&a_{n2}&\ldots&a_{nn}\\\
\end{pmatrix}\begin{pmatrix}y_{1}\\\ y_{2}\\\ \vdots\\\
y_{n}\end{pmatrix},\quad a_{ij}=a_{ij}(x)\in K,\quad y_{i}=y_{i}(x)$ (7)

The fundamental matrix $U=[u_{ij}]$ of the system (7) plays an important role
in this theory because any differential extension of $L/K$ must contains $K$
and $u_{ij}$, i.e., $L=K\langle u_{ij}\rangle$. The Picard-Vessiot extension
$L/K$ is the extension of $K$ preserving the field of constants, that is
$C_{L}=C_{K}$. Thus, given a system of first order linear differential
equations (7), the differential Galois group of $Y^{\prime}=AY$, denoted by
$DGal(L/K)$, is the group of $K$-differential automorphisms from $L$ to $L$,
that is, $\sigma:\,L\mapsto L$, $\partial(\sigma(a))=\sigma(\partial a)$,

$DGal(L/K)=\\{\sigma:\,\,\sigma(a)=a,\,\forall a\in K\\}.$

The connected identity component of $DGal(L/K)$, denoted by $(DGal(L/K))^{0}$
is the biggest algebraic subgroup of $DGal(L/K)$ containing the identity. In a
general framework, more that Hamiltonian systems with homogeneous potentials,
Morales-Ramis theory is the theory that relates differential Galois theory
with the integrability of dynamical systems. In particular, Morales-Ramis
Theorem for Hamiltonian systems says that _if a Hamiltonian system is
integrable, then the connected identity component of the differential Galois
group of the first variational equation along any particular solution is an
abelian group_.

In this way, an _invariant plane_ of the Hamiltonian system with first
integral (3) is a restricted plane satisfying the motion equations. The
solutions of motion equations restricted to the invariant plane are called
_integral particular curves_ , also known _particular solutions_ of the
Hamiltonian system. More precisely, we are interested in the invariant plane
$\Gamma=\\{q_{2}=p_{2}=0\\}$ with particular solution
$\gamma(t)=(q_{1}(t),0,p_{1}(t),0)$. Let be $q_{1}=x$ and
$-V^{\prime}(q_{1},0)=f(x)$. The explicit solutions for differential equation

$\frac{d^{2}x}{dt^{2}}=f(x)$ (8)

are related with the integral curve $(x,\dot{x})$ of the one degree of freedom
Hamiltonian system

$\dot{x}=y,\quad\dot{y}=f(x),$ (9)

where the energy is given by

$h=\frac{y^{2}}{2}-\int_{x_{0}}^{x}f(\tau)d\tau.$ (10)

The first variational equation of the Hamiltonian system (2), along a
particular integral curve $\gamma(t)$ is given by the linear system

$\dot{\mathbf{\xi}}=A(t)\mathbf{\xi},\qquad\mathbf{\xi}=\begin{pmatrix}\xi_{1}\\\
\xi_{2}\\\ \xi_{3}\\\ \xi_{4}\end{pmatrix},\qquad
A(t)=X^{\prime}_{H}\Big{|}_{\gamma(t)},\qquad X_{H}=J_{2}\nabla H,$ (11)

where $X_{H}$ depends on $q_{1},q_{2},p_{1},p_{2}$ and it is restricted to
$\gamma(t)$; $\xi$ is unknown vector which depends on $t$ and $J_{2}$ is the
standard symplectic matrix given by

$J_{2}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\ 0&-1&0&0\end{pmatrix}$

Taking into account the concepts given above, we have the following result.

######  Proposition 2.

Let $k$ be an even number. The Hamiltonian system (2) with exceptional
polynomial potential $V_{k,l}$ has infinite invariant planes

$\Gamma_{\lambda_{1},\lambda_{2}}=\left\\{(q_{1},q_{2},p_{1},p_{2})\in\mathbb{C}^{4}:\,\lambda_{1}q_{1}+\lambda_{2}q_{2}=\lambda_{1}p_{1}+\lambda_{2}p_{2}=0,\,\,\lambda_{1},\lambda_{2}\in\mathbb{C}^{*}\right\\}$
(12)

if and only if $k=2l$ or $\lambda_{1}/\lambda_{2}=\pm i$.

###### Proof.

The system (2) with potential $V_{k,l}$ goes over to

$\begin{array}[]{l}\dot{q}_{1}=p_{1},\\\ \dot{q}_{2}=p_{2},\\\
\dot{p}_{1}=\alpha(q_{2}+iq_{1})^{k-l-1}(q_{2}-iq_{1})^{l-1}(iq_{2}(2l-k)-kq_{1}),\vspace{0.1cm}\\\
\dot{p}_{2}=\alpha(q_{2}+iq_{1})^{k-l-1}(q_{2}-iq_{1})^{l-1}(iq_{1}(k-2l)-kq_{2}).\end{array}$
(13)

Assume $\lambda_{1}\lambda_{2}\neq 0$. The conditions
$\lambda_{1}q_{1}+\lambda_{2}q_{2}=\lambda_{1}p_{1}+\lambda_{2}p_{2}=0$ reads
$q_{2}=cq_{1}$ and $p_{2}=cp_{1}$, where $c=-\lambda_{1}/\lambda_{2}\neq 0$.
Since $(q_{1},cq_{1},p_{1},cp_{1})$ are invariant planes, it is satisfied that
$\dot{q}_{2}=c\dot{q}_{1}$ and $\dot{p}_{2}=cp_{1}$ in Eq. (13). Thus,
$\dot{p}_{2}/\dot{p}_{1}$ in Eq. (13) reads as

$c=\frac{iq_{1}(k-2l)-kq_{2}}{iq_{2}(2l-k)-kq_{1}}=\frac{iq_{1}(k-2l)-ckq_{1}}{icq_{1}(2l-k)-kq_{1}}=\frac{i(k-2l)-ck}{ic(2l-k)-k}.$

Solving the previous expression we obtain $(c^{2}+1)(2l-k)=0$ and due to
$c\neq 0$ we conclude $c=\pm i$ or $k=2l$.

Next, we show that $(q_{1},cq_{1},p_{1},cp_{1})$ is a family of invariant
planes. To see this, we consider $c=\pm i$ in Eq. (13) and trivially
$(q_{1},\pm iq_{1},p_{1},\pm ip_{1})$ are invariant planes. Now, we consider
$k=2l$. Then

$V_{2l,l}=\alpha(q_{1}^{2}+q_{2}^{2})^{l}.$

Since $k=2l$ is an even number, $l\in\mathbb{Z}^{+}$ to avoid the case of
constant polynomial. Hence, the Hamiltonian system associated with this
potential, Eq. (13), becomes

$\begin{array}[]{l}\dot{q}_{1}=p_{1},\\\ \dot{q}_{2}=p_{2},\\\
\dot{p}_{1}=-2l\alpha q_{1}(q_{1}^{2}+q_{2}^{2})^{l-1},\vspace{0.1cm}\\\
\dot{p}_{2}=-2l\alpha q_{2}(q_{1}^{2}+q_{2}^{2})^{l-1}.\end{array}$ (14)

A straightforward analysis of Eq. (14) shows that
$\dfrac{dq_{2}}{dq_{1}}=\dfrac{p_{2}}{p_{1}}$ and
$\dfrac{dp_{2}}{dp_{1}}=\dfrac{q_{2}}{q_{1}}$ imply
$\dfrac{q_{2}}{q_{1}}=\dfrac{p_{2}}{p_{1}}=c\in\mathbb{C}^{*}$. Hence
$c=-\lambda_{2}/\lambda_{1}$,
$\lambda_{1}q_{1}+\lambda_{2}q_{2}=\lambda_{1}p_{1}+\lambda_{2}p_{2}=0$ and
therefore $\Gamma_{\lambda_{1},\lambda_{2}}$ defined in Eq. (12) are infinite
invariant planes of Eq. (13), which completes the proof.

∎

Assuming $\lambda_{1}\lambda_{2}=0$ and $|\lambda_{1}|+|\lambda_{2}|>0$, the
reader should be warned that $(0,q_{2},0,p_{2})$ and $(q_{1},0,p_{1},0)$ are
particular cases of invariant planes $\Gamma_{\lambda_{1},\lambda_{2}}$,
defined in Eq. (12), of the system given in Eq. (13) for $k=2l$. We recall
that $H|_{\Gamma}$ is the hamiltonian restricted to the invariant plane
$\Gamma$. The following result is an application of Morales-Ramis Theory to an
integrable hamiltonian system.

######  Proposition 3.

The connected identity component of the differential Galois group of the first
variational equation of the Hamiltonian system (14), along particular
solutions with $H|_{\Gamma}=0$, is a subgroup of the diagonal group.

###### Proof.

The hamiltonian system (14) reads

$\begin{array}[]{l}\dot{q}_{1}=p_{1},\\\ \dot{q}_{2}=p_{2},\\\
\dot{p}_{1}=-2l\alpha q_{1}(q_{1}^{2}+q_{2}^{2})^{l-1},\vspace{0.1cm}\\\
\dot{p}_{2}=-2l\alpha q_{2}(q_{1}^{2}+q_{2}^{2})^{l-1}.\end{array}$

A particular solution over the invariant plane
$\Gamma=\\{(q_{1},cq_{1},p_{1},cp_{1})\\}$ of Eq. (14) is obtained through the
solution of the differential equation $\dot{p}_{1}=\ddot{q}_{1}=\mu
q_{1}^{2l-1}$, where $\mu=-2l\alpha(c^{2}+1)^{l-1}$. Setting $x=q_{1}$, we
have the differential equation $\ddot{x}=\mu x^{l}$, which corresponds to Eq.
(8) with $f(x)=\mu x^{2l-1}$. Indeed, from (10) we have

$H|_{\Gamma}=h=\frac{(\dot{x})^{2}}{2}-2\nu^{2}x^{2l},\quad-2\nu^{2}=\frac{\mu}{l+1}.$

Owing to $H|_{\Gamma}=h=0$, we obtain the particular solutions

$x_{\pm}(t)=\frac{1}{\sqrt[l-1]{b_{1}\pm b_{2}t}},\qquad
b_{1}=c_{1}(l-1),\qquad b_{2}=2\nu(l-1).$

At this stage we need to obtain the first variational equation along these
particular solutions. It follows that

$\dot{\mathbf{\xi}}=A(t)\mathbf{\xi},\quad\mathbf{\xi}=\begin{pmatrix}\xi_{1}(t)\\\
\xi_{2}(t)\\\ \xi_{3}(t)\\\ \xi_{4}(t)\end{pmatrix},\quad
A(t)=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\
\widehat{\mu}_{1}x(t)^{2l-2}&\widehat{\mu}_{2}x(t)^{2l-2}&0&0\\\
\widehat{\mu}_{2}x(t)^{2l-2}&\widehat{\mu}_{3}x(t)^{2l-2}&0&0\end{pmatrix},$

where $\widehat{\mu}_{1}=-2l\alpha(c^{2}+2l-1)(c^{2}+1)^{l-2}$,
$\widehat{\mu}_{2}=-4l\alpha c(l-1)(c^{2}+1)^{l-2}$ and
$\widehat{\mu}_{3}=-2l\alpha(2lc^{2}-c^{2}+1)(c^{2}+1)^{l-2}$

Since the particular solutions $x_{\pm}(t)$ are algebraic of degree $l-1$, it
follows that

$x_{\pm}(t)^{2l-2}=\frac{1}{\left(b_{1}\pm b_{2}t\right)^{2}},$

and therefore the coefficients matrix of the variational equation has rational
elements. Now, this variational equation has two regular singularities
$t=-b_{1}/b_{2}$ and $t=\infty$, which are of regular type. Moreover, due to
$\widehat{\mu}_{1}\neq\widehat{\mu}_{2}\neq\widehat{\mu}_{3}$ we cannot obtain
logarithmic solutions and the general solution has the form
$c_{1}\xi_{1}(t)+c_{2}\xi_{2}(t)+c_{3}\xi_{3}(t)+c_{4}\xi_{4}(t)$,
$\xi_{k}(t)=g_{k}(t)^{r_{k}}$, where $g_{k}\in\mathbb{C}[t]$ and
$r_{k}\in\mathbb{C}$. Thus, the logarithmic derivative of $\xi_{k}(t)$ is a
rational function and then the connected identity component of the
differential Galois group is contained in the diagonal group. ∎

We observe that considering $h\neq 0$ involves harmonic oscillator ($l=1)$ and
hyperelliptic functions ($l>1)$. For $l=1$ we fall in the case of classical
harmonic oscillator, which is integrable and the variational equations have
constant coefficients, for instance, the differential Galois group is
isomorphic to the multiplicative group, which is a connected group. We cannot
obtain elliptic functions because the exponent is always even ($2l$). At
present we only remark that more information on the relation between
integrability of Hamiltonian systems and solvability of variational equations,
the reader is addressed to [13, 18, 19].

## Acknowledgements

M. Alvarez-Ramírez has been partly supported from Prodep/México grant number
511-6/2019-15963 and the grant Sistemas Hamiltonianos, Mecánica y Geometría
PAPDI2021 CBI-UAMI. P. Acosta-Humánez has been partly supported from Internal
Faculty of Sciences Founds Project No-integrabilidad de campos vectoriales
polinomiales en el plano complejo mediante la Teoría de Galois Diferencial,
Instituto de Matemática. Universidad Autónoma de Santo Domingo.

The authors thank to J. J. Morales-Ruiz for his constructive comments on the
first version of the paper. The authors are indebted with the anonymous
referee by their useful comments and suggestions, which we have taken into
consideration to improve the manuscript.

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