Action-angle variables for the Lie-Poisson Hamiltonian systems associated with Boussinesq equation

Two finite-dimensional Lie-Poisson Hamiltonian systems associated with Boussinesq equation are presented. The action-angle variables in the case of systems with non-hyperelliptic spectral curves are obtained by Sklyanin's method of separation of variables. Moreover, with the help of Hamilton-Jacobi theory for the generating functions of conserved integrals, the Jacobi inversion problems related to the Lie-Poisson Hamiltonian systems and Boussinesq equation are discussed.