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The paper revises the explicit integration of the classical Steklov--Lyapunovsystems via separation of variables, which was first made by F. K\"otter in1900, but was not well understood until recently. We give a geometricinterpretation of the separating variables and then, applying the Weierstrasshyperelliptic root functions, obtain explicit theta-function solution to theproblem. We also analyze the structure of its poles on the correspondingAbelian variety. This enables us to obtain a solution for an alternative set ofphase variables of the systems that has a specific compact form.

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