Painlevé analysis, Prelle–Singer approach, symmetries and integrability of damped Hénon–Heiles system

We consider a modified damped version of Hénon–Heiles system and investigate its integrability. By extending the Painlevé analysis of ordinary differential equations we find that the modified Hénon–Heiles system possesses the Painlevé property for three distinct parametric restrictions. For each of the identified cases, we construct two independent integrals of motion using the well known Prelle–Singer method. We then derive a set of nontrivial non-point symmetries for each of the identified integrable cases of the modified Hénon–Heiles system. We infer that the modified Hénon–Heiles system is integrable for three distinct parametric restrictions. Exact solutions are given explicitly for two integrable cases.