Separation of variables and explicit theta-function solution of the classical Steklov-Lyapunov systems: A geometric and algebraic geometric background

The paper revises the explicit integration of the classical Steklov-Lyapunov systems via separation of variables, which had been first made by F. Kötter in 1900, but was not well understood until recently. We give a geometric interpretation of the separating variables and then, applying the Weierstrass hyperelliptic root functions, obtain explicit theta-function solution to the problem. We also analyze the structure of poles of the solution on the Jacobian on the corresponding hyperelliptic curve. This enables us to obtain a solution for an alternative set of phase variables of the systems that has a specific compact form. In conclusion we discuss the problem of integration of the Rubanovsky gyroscopic generalizations of the above systems.