On bi-Hamiltonian geometry of some integrable systems on the sphere with cubic integral of motion

摘要

We obtain the bi-Hamiltonian structure for a family of integrable systems on the sphere S~2 with an additional integral of third order in momenta. These results are applied to the Goryachev system and Goryachev-Chaplygin top for which we give an explicit procedure to find the separated coordinates and the separated relations.