New variables of separation for the Steklov–Lyapunov system

A rigid body in an ideal fluid is an important example of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(3)=so(3)\ltimes\mathbb R^3$. We present the bi-Hamiltonian structure and the corresponding variables of separation on this phase space for the Steklov–Lyapunov system and it's gyrostatic deformation.