Non-integrability of a three-dimensional generalized Hénon-Heiles system

Abstract

In recent paper Fakkousy et al. show that the 3D Hénon-Heiles system with Hamiltonian $$ H = \frac{1}{2} (p_1 ^2 + p_2 ^2 + p_3 ^2) +\frac{1}{2} (A q_1 ^2 + C q_2 ^2 + B q_3 ^2) + (\alpha q_1 ^2 + \gamma q_2 ^2)q_3 + \frac{\beta }{3}q_3 ^3 $$ is integrable in sense of Liouville when $$\alpha = \gamma , \frac{\alpha }{\beta } = 1, A = B = C$$ ; or $$\alpha = \gamma , \frac{\alpha }{\beta } = \frac{1}{6}, A = C$$ , B-arbitrary; or $$\alpha = \gamma , \frac{\alpha }{\beta } = \frac{1}{16}, A = C, \frac{A}{B} = \frac{1}{16}$$ (and of course, when $$\alpha =\gamma =0$$ , in which case the Hamiltonian is separable). It is known that the second case remains integrable for A, C, B arbitrary. Using Morales-Ramis theory, we prove that there are no other cases of integrability for this system.