Direct Construction of a Bi-Hamiltonian Structure for Cubic Hénon-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 Hénon-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 \(N\).