Darboux integrability of 2-dimensional Hamiltonian systems with homogenous potentials of degree 3

Abstract

We provide a characterization of all Hamiltonian systems of the form \documentclass[12pt]{minimal}\begin{document}$H=(p_1^2\break +p_2^2)/2+ V(q_1,q_2),$\end{document}H=(p12+p22)/2+V(q1,q2), where V is a homogenous polynomial of degree 3 which are completely integrable with Darboux first integrals.