Periodic orbits and equilibria for a seventh-order generalized Hénon-Heiles Hamiltonian system

In this paper we study analytically the existence of two families of periodic orbits using the averaging theory of second order, and the finite and infinite equilibria of a generalized Hénon-Heiles Hamiltonian system which includes the classical Hénon-Heiles Hamiltonian. Moreover we show that this generalized Hénon-Heiles Hamiltonian system is not C¹ integrable in the sense of Liouville–Arnol'd, i.e. it has not a second C¹ first integral independent with the Hamiltonian. The techniques that we use for obtaining analytically the periodic orbits and the non C¹ Liouville-Arnol'd integrability, can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.