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.