Periodic orbits for a generalized Hénon-Heiles Hamiltonian system with an additional singular gravitational term

We study the periodic dynamics of the Hénon-Heiles Hamiltonian system with the additional singular gravitational term 1/(x2+y2). The Hénon-Heiles modelizes how stars move around a galactic center. The addition of this singular gravitational term allows to modelize the motion of the stars in a pseudo or post-Newtonian dynamics. Thus this model allows to predict phenomena which cannot be detected by the classical Newtonian mechanics. Using the averaging theory of first order we study analytically the existence of two families of periodic orbits of this generalized Hénon-Heiles Hamiltonian system. Moreover we characterize when this generalized Hénon-Heiles Hamiltonian system has or has not a second C 1 first integral independent of the Hamiltonian