  We consider a modified damped version of H\'enon-Heiles system and
investigate its integrability. By extending the Painlev\'e analysis of ordinary
differential equations we find that the modified H\'enon-Heiles system
possesses the Painlev\'e 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 H\'enon-Heiles system. We infer that the modified H\'enon-Heiles
system is integrable for three distinct parametric restrictions. Exact
solutions are given explicitly for two integrable cases.
