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

Primitivo B. Acosta-Humánez, Martha Álvarez-Ramírez, Teresinha J. Stuchi

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Primitivo B. Acosta-Humánez, Martha Álvarez-Ramírez, Teresinha J. Stuchi. A note on the integrability of exceptional potentials via polynomial bi-homogeneous potentials. Bull. Comput. Appl. Math., 2021, 9 (2), pp.59-75. hal-03571443

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

Primitivo B. Acosta-Humánez <fnref n="1" /> <fnref n="2" />, Martha Álvarez-Ramírez <fnref n="3" />, Teresinha J. Stuchi <fnref n="4" />

CompAMa Vol.9, No.2, pp.59-75, 2021 - Received April 20, 2021 - Accepted September 15, 2021

Special Issue II JEM 2020, Eds. Carmen Judith Vanegas and Saúl Buitrago Boret

## 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 $l = 0, 1, ..., k$, called exceptional potentials. Hietarinta [1] 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. [2]. 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.

<footnote n="1">¹Instituto de Matemática, Facultad de Ciencias, Universidad Autónoma de Santo Domingo, Dominican Republic. (pacosta-humanez@uasd.edu.do). (ORCID iD: https://orcid.org/0000-0002-5627-4188).</footnote>

<footnote n="2">²Corresponding author.</footnote>

<footnote n="3">³Departamento de Matemáticas, UAM-Iztapalapa, México City, México. (mar@xanum.uam.mx). (ORCID iD: https://orcid.org/0000-0001-9187-1757).</footnote>

<footnote n="4">⁴Instituto de Física, Universidade Federale de Rio de Janeiro, Rio de Janeiro, Brazil. (tstuchi@if.ufjr.br). (ORCID iD: https://orcid.org/0000-0003-0700-6205).</footnote>
## Resumen

Este artículo trata de la integrabilidad polinomial de sistemas hamiltonianos potenciales polinomiales homogéneos 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 [1] 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. [2]. 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 [3]. 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 [4, 5] was started by Yoshida [6, 7]. 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 [8–10] and references therein.

We assume that $\mathbb{C}^{2n}$ is the ymplectic 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

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

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

where $V'(\mathbf{q})$ 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 $V'(\mathbf{q}) = \lambda\mathbf{d}$, 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 $V'(\mathbf{d}) \neq 0$. 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 $\mathbb{CP}^{n-1}$ with $[d_1 : \dots : d_n]$ as homogeneous coordinates. For more information on this subject the reader is referred to [9].

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 $A_{2 \times 2} \in \mathrm{PO}(2, \mathbb{C}) \cong \mathbb{CP}^1$. Here $\mathrm{PO}(2, \mathbb{C})$ 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 $A \in \mathrm{PO}(2, \mathbb{C})$ such that $V_1(\mathbf{q}) = V_2(A\mathbf{q})$. This concept was inspired by a Hietarinta's result proved in [11], namely that if $V_1(\mathbf{q})$ is integrable, then $V_2(A\mathbf{q})$ is also integrable for any $A \in \mathrm{PO}(2, \mathbb{C})$.

A homogeneous polynomial $V \in \mathbb{C}[q_1, q_2]$ of degree $k$ is a polynomial such that $V(q_1, q_2) = q_1^k V(1, q_2/q_1) = q_2^k V(q_1/q_2, 1)$. Let be $v(x) := V(1, x)$ and $w(\xi) := V(\xi, 1)$. Since $\mathrm{PO}(2, \mathbb{C}) = \mathbb{CP}^1$, we use $[q_1 : q_2]$ as homogeneous coordinates on $\mathbb{CP}^1$. Moreover, points $[1 : q_2] \in \mathbb{CP}^1$ will be identified with the affine part of $\mathbb{CP}^1$ 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 [9], the authors choose amid all equivalent points a representative $V$, whose polynomial $w$ (or $v$) has one zero at some point of $\mathbb{CP}^1 \setminus \{[1 : +i], [1 : -i]\}$. This always happens, except for the named *exceptional potentials* given by

$$ V_{k,l}(q_1, q_2) = \alpha(q_2 - iq_1)^l(q_2 + iq_1)^{k-l}, \text{ where } l = 0, \dots, k \text{ and } \alpha \in \mathbb{C} \setminus \{0\}. $$

Indeed, it follows that these potentials lack Darboux points, see [9]. 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

$$H = \frac{1}{2}(p_1^2 + p_2^2) + V_{k,l}(q_1, q_2), \qquad (3)$$

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

$$V_{k,l}(q_1, q_2) = \alpha(q_2 - iq_1)^l(q_2 + iq_1)^{k-l}, \quad l = 0, 1, \dots, k, \qquad (4)$$

where $k$ is a positive integer. For more details, see [9].

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 [3].

In [1], 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 $k \le 4$ are completely integrable with a polynomial integral besides the classical first integral. Additionally, Llibre and Valls in [12] proved that Hamiltonian systems having potential $V_{k,l}$ with even $k \ge 6$ 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 [12] 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. [13] 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 [14] claimed that by using the direct method they have checked that for $5 \le k \le 20$, $k \ne 7$, the Hamiltonian system with two degree of freedom and potential $V_{k,l}$ and $l \notin \{0, 1, k/2\}$, 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 [2], Combot et al. used a theorem by Moulin Ollagnier about integrability of the homogeneous Lotka-Volterra system (see [15]), 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

$$H = 2y_1y_2 + \beta x_1^l x_2^{k-l}, \qquad (5)$$

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

$$V_{k,l} = \beta x_1^l x_2^{k-l}, \qquad (6)$$

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 [2]). *The Hamiltonian system* $H(x_1, x_2, y_1, y_2) = 2y_1y_2 + x_1^{k_1}x_2^{k_2}$ 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

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

**Theorem 2** (Combot et al. [2]). *The Hamiltonian system given by* $H(x_1, x_2, y_1, y_2) = 2y_1y_2 + x_1^{k_1}x_2^{k_2}$ with $(k_1, k_2) \in \mathbb{Q}^2$ is algebraically integrable if and only if

1. $(k_1, k_2) = (0, k)$, the additional first integral being $J = y_1$.

2. $(k_1, k_2) = (k, k)$, $J = y_1x_1 - y_2x_2$.
3. $(k_1, k_2) = (-\frac{1}{2}, k)$, $J = y_1(y_1x_1 - y_2x_2) - \frac{1}{2}x_1^{-1/2}x_2^{k+1}$.

4. $(k_1, k_2) = (1, k)$, $J = y_1^2 + \frac{1}{k+1}x_2^{k+1}$.

5. $(k_1, k_2) = (-k-2, k)$, $J = (y_1x_1 - y_2x_2)^2 - 2x_1^{-k-1}x_2^{k+1}$.

6. $(k_1, k_2) = (2, 5)$, $J = 6y_1^3(y_1x_1 - y_2x_2) + x_2^6(4y_1^2x_1^2 - 8y_1y_2x_1x_2 + y_2^2x_2^2) - x_1^3x_2^{12}$.

7. $(k_1, k_2) = (2, -7/4)$,

$$J = y_1^6 + 8x_2^{-3/4}y_1^3(4y_2x_2 - y_1x_1) + 8x_2^{-3/2}(5y_1^2x_1^2 - 4y_1y_2x_1x_2 + 8y_2^2x_2^2) - 48x_1^3x_2^{-9/4},$$

8. $(k_1, k_2) = (2, -10/7)$,

$$J = 4y_1^6 + 14x_2^{-3/7}y_1^3(7y_2x_2 - 4y_1x_1) + 343x_2^{-6/7}(y_1x_1 - 7y_2x_2)(y_1x_1 - y_2x_2) - 2058x_1^3x_2^{-9/7},$$

9. $(k_1, k_2) = (3, -9/5)$,

$$J = 4y_1^4 - 10x_2^{-4/5}(3y_1^2x_1^2 - 30y_1y_2x_1x_2 + 25y_2^2x_2^2) + 225x_1^4x_2^{-8/5},$$

10. $(k_1, k_2) = (-2/3, -7/3)$,

$$J = 8y_1(y_1x_1 - y_2x_2)^3 - x_1^{-2/3}x_2^{-4/3}(13y_1^2x_1^2 - 44y_1y_2x_1x_2 + 4y_2^2x_2^2) + 16x_1^{-1/3}x_2^{-8/3},$$

11. $(k_1, k_2) = (-2/3, -10/3)$,

$$J = 4y_1^2(y_1x_1 - y_2x_2)^4 + 2x_1^{-2/3}x_2^{-7/3}y_1(-4y_1^3x_1^3 + 13y_1^2y_2x_1^2x_2 - 20y_1y_2^2x_1x_2^2 + 2y_2^3x_2^3),$$

12. $(k_1, k_2) = (-2/3, -5/6)$,

$$J = 2x_1^{-2/3}(y_1x_1 - y_2x_2)^2[x_2^{1/6} - 2y_1x_1^{2/3}(y_1x_1 - y_2x_2)] + x_1^{-1/3}x_2^{1/3},$$
$$13. (k_1, k_2) = (-3/4, -9/4),$$

$$J = 4y_1(y_1x_1 - y_2x_2)^3 - 2x_1^{-3/4}x_2^{-5/4}(3y_1^2x_1^2 - 12y_1y_2x_1x_2 + y_2^2x_2^2) + 9x_1^{-1/2}x_2^{-5/2}.$$

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

$$1. V_{k,0} = \beta x_2^k, \quad \beta = i^k \alpha, \quad J = y_1.$$

$$2. V_{k,k} = \beta x_1^k, \quad \beta = (-i)^k \alpha, \quad J = y_2.$$

$$3. V_{k,k/2} = \beta x_1^{k/2} x_2^{k/2}, \quad \text{with even } k, \beta = \alpha, \quad J = y_1 x_1 - y_2 x_2.$$

$$4. V_{k,1} = \beta x_1 x_2^{k-1}, \quad \beta = -i^k \alpha, \quad J = y_1^2 - \frac{1}{k} i^k \alpha x_2^k.$$

$$5. V_{k,k-1} = \beta x_1^{k-1} x_2, \quad \beta = (-1)^{k-1} i^k \alpha, \quad J = y_2^2 + \frac{1}{k} (-1)^{k+1} i^k \alpha x_1^k.$$

$$6. V_{7,2} = \beta x_1^2 x_2^5, \quad \beta = -i\alpha, \quad \alpha = i, \\ J = 16y_1^3(y_1x_1 - y_2x_2) + x_2^6(4y_1^2x_1^2 - 8y_1y_2x_1x_2 + y_2^2x_2^2) - x_1^3x_2^{12}.$$

$$7. V_{7,5} = \beta x_1^5 x_2^2, \quad \beta = i\alpha, \quad \alpha = -i, \\ J = 16y_2^3(y_2x_2 - y_1x_1) + x_1^6(4y_2^2x_2^2 - 8y_2y_1x_1x_2 + y_1^2x_1^2) - x_2^3x_1^{12}.$$

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

$$H(x_1, x_2, y_1, y_2) = 2y_1y_2 + V_{k,l}(x_1, x_2),$$

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. Then, 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. $V_{k,0} = i^k \alpha x_2^k, \quad J = y_1$
$$ \{H, J\} = 2y_2 \cdot 0 + 2y_1 \cdot 0 + 0 \cdot 1 - ki^k \alpha x_2 k - 1 \cdot 0 = 0. $$

2. $V_{k,k} = (-i)^k \alpha x_1^k, \quad J = y_2$
$$ \{H, J\} = 2y_2 \cdot 0 + (-1)^{k+1} ki^k \alpha x_1 k - 1 \cdot 0 + 2y_1 \cdot 0 + 0 \cdot 1 = 0. $$

3. $V_{k,k/2} = \alpha x_1^{k/2} x_2^{k/2}, \quad J = y_1 x_1 - y_2 x_2$
$$ \{H, J\} = 2y_2 y_1 - \frac{1}{2} k \alpha x_1^{\frac{k}{2}-1} x_2^{k/2} \cdot x_1 - 2y_1 y_2 - \frac{1}{2} k \alpha x_1^{k/2} x_2^{\frac{k}{2}-1} \cdot (-x_2) = 0. $$

4. $V_{k,1} = -i^k \alpha x_1 x_2^{k-1}, \quad J = y_1^2 - \frac{1}{k} i^k \alpha x_2^k$
$$ \begin{aligned} \{H, J\} &= 2y_2 \cdot 0 + i^k \alpha x_2^{k-1} \cdot (2y_1) + 2y_1 \cdot (-i^k \alpha x_2^{k-1}) \\ & \quad + i^k (k-1) \alpha x_1 x_2^{k-2} \cdot 0 \\ &= 2i^k \alpha x_2^{k-1} y_1 - 2i^k \alpha x_2^{k-1} y_1 = 0 \end{aligned} $$

5. $V_{k,k-1} = (-1)^{k-1} i^k \alpha x_1^{k-1} x_2, \quad J = y_2^2 + \frac{1}{k} (-1)^{k+1} i^k \alpha x_1^k$
$$ \begin{aligned} \{H, J\} &= 2y_2 \cdot (-1)^{k+1} i^k \alpha x_1^{k-1} + (-1)^k i^k (k-1) \alpha x_1^{k-2} x_2 \cdot 0 + 2y_1 \cdot 0 \\ & \quad + (-1)^k i^k \alpha x_1^{k-1} \cdot (2y_2) \\ &= 0 \end{aligned} $$

6. $V_{7,2} = x_1^2 x_2^5, \quad J = 16y_1^3(y_1 x_1 - y_2 x_2) + x_2^6(4y_1^2 x_1^2 - 8y_1 y_2 x_1 x_2 + y_2^2 x_2^2) - x_1^3 x_2^{12}$

$$
\begin{aligned} \{H, J\} &= -6x_1^2 x_2^{12} y_2 + 16x_1 x_2^6 y_1^2 y_2 + 32y_1^4 y_2 - 16x_2^7 y_1 y_2^2 \\ & \quad - 16x_1^3 x_2^{11} y_1 - 128x_1^2 x_2^5 y_1^3 + 16x_1^2 x_2^{12} y_2 + 96x_1 x_2^6 y_1^2 y_2 \\ & \quad - 24x_1^3 x_2^{11} y_1 + 48x_1^2 x_2^5 y_1^3 - 112x_1 x_2^6 y_1^2 y_2 - 32y_1^4 y_2 \\ & \quad + 16x_2^7 y_1 y_2^2 + 40x_1^3 x_1^{11} y_1 + 80x_1^2 x_2^5 y_1^3 - 10x_1^2 x_2^{12} y_2 \\ & = 0 \end{aligned}
$$
$$7. V_{7,5} = x_1^5 x_2^2,$$

$$J = 16y_2^3(y_2x_2 - y_1x_1) + x_1^6(4y_2^2x_2^2 - 8y_2y_1x_1x_2 + y_1^2x_1^2) - x_2^3x_1^{12}$$

$$\{H, J\} = 2y_2 \cdot (-4)(3x_1^{11}x_2^3 - 2x_1^7y_1^2 + 14x_1^6x_2y_1y_2 - 6x_1^5x_2^2y_2^2 + 4y_1y_2^3 \\ - 5x_1^4x_2^2 \cdot 2(x_1^8y_1 - 4x_1^7x_2y_2 - 8x_1y_2^3) \\ + 2y_1 \cdot (-3x_1^{12}x_2^2 - 8x_1^7y_1y_2 + 8x_1^6x_2y_2^2 + 16y_2^4) \\ - 2x_1^5x_2 \cdot 8(-x_1^7x_2y_1 + x_1^6x_2^2y_2 - 6x_1y_1y_2^2 + 8x_2y_2^3) = 0$$

□

**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.

Let us conclude by saying that our results agree well with those of Hietarinta [11] and Nakagawa et al. [13]. 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
<table><caption>Tab. 1: Exceptional potential $V_{k,l}$ and the additional first integral $J$ in terms of their natural variables $(q_1, q_2, p_1, p_2)$.</caption><thead><tr><th>Exceptional potential V<sub>k,l</sub></th><th>First integral J</th></tr></thead><tbody><tr><td>V<sub>k,0</sub> = α(q<sub>2</sub> + iq<sub>1</sub>)<sup>k</sup></td><td><sup>1</sup>&frasl;<sub>2</sub>(p<sub>1</sub> - ip<sub>2</sub>)</td></tr><tr><td>V<sub>k,k</sub> = α(q<sub>2</sub> - iq<sub>1</sub>)<sup>k</sup></td><td><sup>1</sup>&frasl;<sub>2</sub>(p<sub>1</sub> + ip<sub>2</sub>)</td></tr><tr><td>V<sub>k,k/2</sub> = α(q<sub>2</sub> - iq<sub>1</sub>)<sup>k/2</sup>(q<sub>2</sub> + iq<sub>1</sub>)<sup>k/2</sup></td><td>-<sup>1</sup>&frasl;<sub>2</sub>(p<sub>1</sub>q<sub>2</sub> - q<sub>1</sub>p<sub>2</sub>)</td></tr><tr><td>V<sub>k,1</sub> = α(q<sub>2</sub> - iq<sub>1</sub>)(q<sub>2</sub> + iq<sub>1</sub>)<sup>k-1</sup></td><td><sup>1</sup>&frasl;<sub>4</sub>(p<sub>1</sub> - ip<sub>2</sub>)<sup>2</sup> - <sup>1</sup>&frasl;<sub>k</sub>i<sup>k</sup>α(q<sub>1</sub> - iq<sub>2</sub>)<sup>k</sup></td></tr><tr><td>V<sub>k,k-1</sub> = α(q<sub>2</sub> - iq<sub>1</sub>)<sup>k-1</sup>(q<sub>2</sub> + iq<sub>1</sub>)</td><td><sup>1</sup>&frasl;<sub>4</sub>(p<sub>1</sub> + ip<sub>2</sub>)<sup>2</sup> + <sup>1</sup>&frasl;<sub>k</sub>(-1)<sup>k+1</sup>i<sup>k</sup>α(q<sub>1</sub> + iq<sub>2</sub>)<sup>k</sup></td></tr><tr><td>V<sub>7,2</sub> = (q<sub>1</sub> - iq<sub>2</sub>)<sup>5</sup>(q<sub>1</sub> + iq<sub>2</sub>)<sup>2</sup></td><td>- (p<sub>1</sub> - ip<sub>2</sub>)<sup>3</sup>(p<sub>1</sub> + ip<sub>2</sub>)(q<sub>1</sub> - iq<sub>2</sub>) + <sup>1</sup>&frasl;<sub>4</sub>(p<sub>1</sub> + ip<sub>2</sub>)<sup>2</sup>(q<sub>1</sub> - iq<sub>2</sub>)<sup>8</sup><br/>+ (p<sub>1</sub> - ip<sub>2</sub>)<sup>4</sup>(q<sub>1</sub> + iq<sub>2</sub>) - 2(p<sub>1</sub> - ip<sub>2</sub>)(p<sub>1</sub> + ip<sub>2</sub>)(q<sub>1</sub> - iq<sub>2</sub>)<sup>7</sup>(q<sub>1</sub> + iq<sub>2</sub>)<br/>+ (p<sub>1</sub> - ip<sub>2</sub>)<sup>2</sup>(q<sub>1</sub> - iq<sub>2</sub>)<sup>6</sup>(q<sub>1</sub> + iq<sub>2</sub>)<sup>2</sup> - (q<sub>1</sub> - iq<sub>2</sub>)<sup>12</sup>(q<sub>1</sub> + iq<sub>2</sub>)<sup>3</sup></td></tr><tr><td>V<sub>7,5</sub> = (q<sub>1</sub> - iq<sub>2</sub>)<sup>2</sup>(q<sub>1</sub> + iq<sub>2</sub>)<sup>5</sup></td><td>(p<sub>1</sub> + ip<sub>2</sub>)<sup>4</sup>(q<sub>1</sub> - iq<sub>2</sub>) - (p<sub>1</sub> - ip<sub>2</sub>)(p<sub>1</sub> + ip<sub>2</sub>)<sup>3</sup>(q<sub>1</sub> + iq<sub>2</sub>)<br/>+ (p<sub>1</sub> + ip<sub>2</sub>)<sup>2</sup>(q<sub>1</sub> - iq<sub>2</sub>)<sup>2</sup>(q<sub>1</sub> + iq<sub>2</sub>)<sup>6</sup><br/>-2(p<sub>1</sub> - ip<sub>2</sub>)(p<sub>1</sub> + ip<sub>2</sub>)(q<sub>1</sub> - iq<sub>2</sub>)(q<sub>1</sub> + iq<sub>2</sub>)<sup>7</sup><br/>+ <sup>1</sup>&frasl;<sub>4</sub>(p<sub>1</sub> - ip<sub>2</sub>)<sup>2</sup>(q<sub>1</sub> + iq<sub>2</sub>)<sup>8</sup> - (q<sub>1</sub> - iq<sub>2</sub>)<sup>3</sup>(q<sub>1</sub> + iq<sub>2</sub>)<sup>12</sup></td></tr></tbody></table>

first variational equation. For more information on this subject the reader is referred to [16–21].

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:

$$1. \partial(a + b) = \partial a + \partial b$$

$$2. \partial(a \cdot b) = a \cdot \partial b + \partial a \cdot b$$

$$3. \partial\left(\frac{a}{b}\right) = \frac{1}{b^2}(a \cdot \partial b - \partial a \cdot b).$$

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

$$\frac{d}{dx} \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, \quad a_{ij} = a_{ij}(x) \in K, \quad y_i = y_i(x) \qquad (7)$$
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 $Y' = AY$, denoted by $DGal(L/K)$, is the group of $K$-differential automorphisms from $L$ to $L$, that is, $\sigma: L \to L$, $\partial(\sigma(a)) = \sigma(\partial a)$,

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

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 $-V'(q_1, 0) = f(x)$. The explicit solutions for differential equation

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

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

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

where the energy is given by

$$h = \frac{y^2}{2} - \int_{x_0}^{x} f(\tau) d\tau. \qquad (10)$$
The first variational equation of the Hamiltonian system (2), along a particular integral curve $\gamma(t)$ is given by the linear system

$$ \dot{\xi} = A(t)\xi, \quad \xi = \begin{pmatrix} \xi_1 \\ \xi_2 \\ \xi_3 \\ \xi_4 \end{pmatrix}, \quad A(t) = X'_H|_{\gamma(t)}, \quad X_H = J_2\nabla H, \quad (11) $$

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

$$ \Gamma_{\lambda_1, \lambda_2} = \{(q_1, q_2, p_1, p_2) \in \mathbb{C}^4 : \lambda_1 q_1 + \lambda_2 q_2 = \lambda_1 p_1 + \lambda_2 p_2 = 0, \lambda_1, \lambda_2 \in \mathbb{C}^*\} \quad (12) $$

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

$$ \begin{align} \dot{q}_1 &= p_1, \notag \\ \dot{q}_2 &= p_2, \notag \\ \dot{p}_1 &= \alpha(q_2 + iq_1)^{k-l-1}(q_2 - iq_1)^{l-1}(iq_2(2l-k) - kq_1), \tag{13} \\ \dot{p}_2 &= \alpha(q_2 + iq_1)^{k-l-1}(q_2 - iq_1)^{l-1}(iq_1(k-2l) - kq_2). \notag \end{align} $$

Assume $\lambda_1\lambda_2 \neq 0$. The conditions $\lambda_1q_1 + \lambda_2q_2 = \lambda_1p_1 + \lambda_2p_2 = 0$ 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

$$ \begin{aligned} \dot{q}_1 &= p_1, \\ \dot{q}_2 &= p_2, \\ \dot{p}_1 &= -2l\alpha q_1(q_1^2 + q_2^2)^{l-1}, \\ \dot{p}_2 &= -2l\alpha q_2(q_1^2 + q_2^2)^{l-1}. \end{aligned} \qquad (14) $$

A straightforward analysis of Eq. (14) shows that $\frac{dq_2}{dq_1} = \frac{p_2}{p_1}$ and $\frac{dp_2}{dp_1} = \frac{q_2}{q_1}$ imply $\frac{q_2}{q_1} = \frac{p_2}{p_1} = c \in \mathbb{C}^*$. 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. $\square$

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

$$ \begin{aligned} \dot{q}_1 &= p_1, \\ \dot{q}_2 &= p_2, \\ \dot{p}_1 &= -2l\alpha q_1(q_1^2 + q_2^2)^{l-1}, \\ \dot{p}_2 &= -2l\alpha q_2(q_1^2 + q_2^2)^{l-1}. \end{aligned} $$
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

$$x_{\pm}(t) = \frac{1}{\sqrt[l-1]{b_1 \pm b_2 t}}, \quad b_1 = c_1(l-1), \quad b_2 = 2\nu(l-1).$$

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

$$\dot{\xi} = A(t)\xi, \quad \xi = \begin{pmatrix} \xi_1(t) \\ \xi_2(t) \\ \xi_3(t) \\ \xi_4(t) \end{pmatrix}, \quad A(t) = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \hat{\mu}_1 x(t)^{2l-2} & \hat{\mu}_2 x(t)^{2l-2} & 0 & 0 \\ \hat{\mu}_2 x(t)^{2l-2} & \hat{\mu}_3 x(t)^{2l-2} & 0 & 0 \end{pmatrix},$$

where $\hat{\mu}_1 = -2l\alpha(c^2 + 2l - 1)(c^2 + 1)^{l-2}$, $\hat{\mu}_2 = -4l\alpha c(l-1)(c^2 + 1)^{l-2}$ and $\hat{\mu}_3 = -2l\alpha(2lc^2 - c^2 + 1)(c^2 + 1)^{l-2}$

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

$$x_{\pm}(t)^{2l-2} = \frac{1}{(b_1 \pm b_2 t)^2},$$

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 $\hat{\mu}_1 \neq \hat{\mu}_2 \neq \hat{\mu}_3$ 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. $\square$

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 [6–8].

Acknowledgment

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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