The Hénon-Heiles system as part of an integrable system in five unknowns with three constants of motions

Abstract: In this paper we construct a new completely integrable system. This system is an instance of a master system of differential equations in $5$ unknowns having $3$ quartics constants of motion.We find via the Painlev\'e analysis the principal balances of the hamiltonian field defined by the hamiltonian. Consequently, the system in question is algebraically integrable. A careful analysis of this system reveals an intimate rational relationship with a special case of the well known H\'{e}non-Heiles system. The latter admits asymptotic solutions with fractional powers in $t$ and depending on $3$ free parameters. As a consequence, this system is algebraically completely integrable in the generalized sense.