A proper surface of section to study a new Hénon–Heiles potential with additional singular gravitational terms

The analysis of the escape of a test particle from a Henon–Heiles potential requires the computation of the ingoing and outgoing asymptotic trajectories to the periodic orbits located at the openings of the potential well. This analysis must be performed in an adequate surface of section. The presence of an additional gravitational potential, including a singularity at the origin of the system, makes impossible to carry out this study using the classical surfaces of section. The aim of this article is to contribute a new surface of section that allows the analysis of the sequence of intersections of those asymptotic trajectories to the Lyapunov orbits with the surface of section in a very simple way.