The generalized Hénon-Heiles system, abelian surfaces and algebraic complete integrability

Abstract

The paper deals with a systematic geometric approach to integration of generalized Hénon-Heiles system. This problem is defined by a nonlinear system of four differential equations admitting two polynomial first integrals. We prove that the problem can be linearized, i.e., it can be written in terms of Abelian integrals, on the dual of a Prym variety Prym * σ( D) of a smooth genus three hyperelliptic Riemann surface D.