Separation of variables for the Clebsch model: so(4) spectral/separation curves

Abstract

In the present paper we construct symmetric separation of variables (SoV) for the anisotropic Clebsch model for which both curves of separation coincide with spectral curve $K$ of its $so\left(4\right)$-valued Lax matrix. We explicitly construct coordinates and momenta of separation, Abel-type quadratures and reconstruction formulae for the presented new SoV. The found SoV is the first example of SoV for integrable systems with generic $so\left(n\right)$-valued Lax matrices when all curves of separation coincide with a spectral curve of the corresponding $so\left(n\right)$-valued Lax matrix and $n>3$.