We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the two samples of Lie algebra e(3). Using this map we establish equivalence of the Steklov-Lyapunov system and the motion of a particle on the surface of the sphere under the influence of the fourth order potential. To study separation of variables for the Steklov case on the Lie algebra so(4) we use the twisted Poisson map between the bi-Hamiltonian manifolds e(3) and so(4).