Generalized Lenard chains and multi-separability of the Smorodinsky–Winternitz system

We show that the notion of generalized Lenard chains allows to formulate in a natural way the theory of multi–separable systems in the context of bi–Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function and by a Nijenhuis tensor defined on a symplectic manifold guarantees the separation of variables. As an application, we construct such a chain for the case I of the classical Smorodinsky–Winternitz model.