Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields   of the circle and periodic shallow water equations
  This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. We investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures called affine or modified Lie-Poisson structures are involved in the integrability of certain Euler equations that arise as models of shallow water waves. 
Boris Kolev (LATP)