Canonical separating coordinates in the generalized cubic H\'enon-Heiles systems
We study the three classical integrable generalized cubic H\'enon--Heiles systems -- Kaup--Kupershmidt, KdV$_5$, and Sawada--Kotera -- from the viewpoint of bi-Hamiltonian geometry and separation of variables. On the standard symplectic manifold $T^*\mathbb R^2$, we construct compatible Poisson deformations $P_1=L_XP_0$, compute the associated recursion operators $N=P_1P_0^{-1}$, and analyze the action of $N^*$ on the codistribution generated by the first integrals. This yields the corresponding control matrices, whose eigenvalues provide the separating coordinates. For the generalized Kaup--Kupershmidt case we carry out the construction explicitly: we determine a deformation vector field, the compatible Poisson tensor, the torsionless recursion operator, the control matrix, the separating coordinates, and, crucially, the conjugate momenta. We then derive the separated relations and write the Hamilton equations in canonical separated variables, thus decomposing the original Hamiltonian system into two separated subsystems. To the best of our knowledge, this explicit derivation of the separating variables and, in particular, of the conjugate momenta for the generalized Kaup--Kupershmidt system is new. For the KdV$_5$ and Sawada--Kotera cases we show how the same bi-Hamiltonian scheme applies, emphasizing both the common geometric mechanism and the features peculiar to each system. In this way, the three generalized cubic H\'enon--Heiles systems are treated within a unified framework based on compatible Poisson structures, recursion operators, control matrices, and Darboux--Nijenhuis coordinates.