Orbits in the integrable H\'enon-Heiles systems

We study in detail the form of the orbits in integrable generalized H\'enon-Heiles systems with Hamiltonians of the form $H = \frac{1}{2}(\dot{x}^2 + Ax^2 + \dot{y}^2 + By^2) + \epsilon(xy^2 + \alpha x^3).$ In particular, we focus on the invariant curves on Poincar\'e surfaces of section ($y = 0$) and the corresponding orbits on the $x-y$ plane. We provide a detailed analysis of the transition from bounded to escaping orbits in each integrable system case, highlighting the mechanism behind the escape to infinity. Then, we investigate the form of the non-escaping orbits, conducting a comparative analysis across various integrable cases and physical parameters.