On the bi-Hamiltonian structure of the Goryachev system on the sphere

A second Poisson bivector consistent with the canonical bivector in the framework of bi-Hamiltonian geometry is constructed and the separation variables for the Goryachev system with an integral of the third order in momenta are found. The Goryachev system is described by using the standard coordinates that are two-dimensional vectors on the Poisson manifold. For the Goryachev system, the second Poisson brackets on the algebra have a certain form, while the bi-involution condition for the integrals implies that there exists a nonsingular matrix. The results also show that the Goryachev system has a more general solution in the framework of the polynomial ansatz proposed for the vector field, leading to the separation variables and separated equations.