Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation   variables
  We discuss from a bi-Hamiltonian point of view the Hamilton-Jacobi separability of a few dynamical systems. They are shown to admit, in their natural phase space, a quasi-bi-Hamiltonian formulation of Pfaffian type. This property allows us to straightforwardly recover a set of separation variables for the corresponding Hamilton-Jacobi equation. 
G. Tondo, C. Morosi