Eur. Phys. J. Plus (2021) 136: 1039
https://doi.org/10.1140/epjp/s13360-021-02044-0

Regular Article

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

Faculty of Mathematics and Informatics, Sofia University, 5 J. Bouchier Blvd., 1164, Sofia, Bulgaria

^a christov@fmi.uni-sofia.bg

Received: 23 March 2021
Accepted: 5 October 2021
Published online: 15 October 2021

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.