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

Abstract: In paper Fakkousy et al. show that the 3D H\'{e}non-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.