DIRECT CONSTRUCTION OF A BI-HAMILTONIAN STRUCTURE FOR CUBIC HENON-HEILES SYSTEMS

The problem of separating variables in integrable Hamiltonian systems has been extensively studied in the last decades. A recent approach is based on the so called Kowalewski's Conditions used to characterized a Control Matrix M whose eigenvalues give the desired coordinates. In this paper we calculate directly a second compatible Hamiltonian structure for the cubic Henon-Heiles systems and in this way we obtain the separation variables as the eigenvalues of a recursion operator N. Finally we re-obtain the Control Matrix M from N.