Separation of variables for the classical elliptic reflection equation algebra

Abstract

In the present paper we construct separation of variables (SoV) for all Lax-integrable systems, two by two Lax matrix of which enjoys classical reflection equation algebra with the elliptic r−s matrices. We show that, similar to the cases of SoV for the classical XYZ and XXZ models [1,2], the constructed SoV admits two types of momenta, which are important in the quantum case [1,2]. We consider two examples of such the integrable hamiltonian systems governed by the simplest Poisson brackets connected with classical reflection equation algebra with the elliptic r−s matrices. They are the classical Sklyanin algebra and its four-dimensional extension which provide a quadratic structure for the classical Steklov top. We explicitly construct the variables of separation, Abel equations and reconstruction formulae for the corresponding integrable models.