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.

Comments are welcome! arXiv admin note: text overlap with arXiv:1503.08171; text overlap with arXiv:2006.15908 by other authors