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 $N$..
"Direct Construction of a Bi-Hamiltonian Structure for Cubic Hénon-Heiles Systems." J. Geom. Symmetry Phys. 57 99 - 109, 2020. https://doi.org/10.7546/jgsp-57-2020-99-109