Bihamiltonian structures and Stäckel separability

It is shown that a class of Stäckel separable systems is characterized in terms of a Gel’fand–Zakharevich bihamiltonian structure. This structure arises as an extension of a Poisson–Nijenhuis structure on phase space. It is also shown that the Casimir of the Gel’fand–Zakharevich bihamiltonian structure provides the family of commuting Killing tensors found by Benenti and that, because of Eisenhart’s theorem, characterize orthogonal separability. It is also shown that recently found properties of quasi-bihamiltonian systems are natural consequences of the geometry of the extension of the Poisson–Nijenhuis structure.