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

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 C and l=0,1,..., 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.

Keywords

• hamiltonian system with two degrees of freedom
• homogeneous potentials
• integrability