On the Separation of Variables in Bi-Hamiltonian Systems

A bi-Hamiltonian system is a Hamiltonian system which allows Hamiltonian formulations by means of two different, but compatible Poisson brackets. The two Poisson brackets allow to construct a number of integrals in involution, which is an interesting and important relation between bi-Hamiltonian systems and integrability. Based on the paper 'On The Local Geometry Of A Bihamiltonian Structure' by Gelfand and Zakharevich, this talk focuses on the aspect of separation of variables in bi-Hamiltonian systems.  We will formulate and prove a theorem to claim that, under certain assumptions, the separation of variables in bi-Hamiltonian systems follows the eigendistribution of the recursion operator.