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

TL;DR

This paper investigates the polynomial integrability of certain two-dimensional Hamiltonian systems with exceptional polynomial potentials, providing an elementary proof and discovering a new first integral for a specific case.

Contribution

It offers a simplified proof of known integrability results and introduces a new first integral for the potential V_{7,5}.

Findings

Elementary proof of polynomial integrability for specific potentials.

Discovery of a new first integral for V_{7,5}.

Galoisian analysis for the case l=k/2.

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 $V_{k,l}=\alpha (q_2-i q_1)^l (q_2+iq_1)^{k-l}$ with $\alpha\in\mathbb{C}$ and $l=0,1,\dots, k$, called exceptional potentials. Hietarinta \cite{Hietarinta1983} 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. \cite{Combot2020}. 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 $l=k/2$.