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 was first made by F. K\"otter 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 its poles on the corresponding
Abelian variety. This enables us to obtain a solution for an alternative set of
phase variables of the systems that has a specific compact form.Comment: 21 pages, 4 figure
