Abstract The paper revises the separation of variables and explicit integration of the classical Steklov{ Lyapunov systems, which was ¯rst made by F. KÄotter in 1900. Namely, we give a geometric interpretation of the separating variables and, then, applying the Weierstrass root functions, obtain an explicit theta-function solution to the problem. We also give such a solution for an alternative set of phase variables of the systems that has a much simpler form.