Algebraic Geometric Poisson Brackets in the Problem of Exact Integration

When solving the problem on finding exact solution of an integrable Hamiltonian system, one usually choose a mapping (covering) that transforms the original system into a system of Abel equations determined in a space of hyperelliptic bundles. The analytical Poisson bracket is induced in this space. In the paper we show that the Jacobi identity imposes certain conditions on the polynomial and constants, which enter the system of Abel equations. This fact allows us to calculate the corresponding constants and to find action variables in the Steklov-Lyapunov problem and, consequently, to complete exact integration of this problem.