Symmetric separation of variables, shifted spectral curves and classical r-matrices

Abstract

We develop a method of separating functions in the theory of variable separation for the Lax-integrable hamiltonian systems. For the case of $gl\left(2\right)$-valued Lax matrices we propose a modified approach to construction of separating functions leading to shifted spectral curves of the initial Lax matrices. In particular, we construct one-parametric families of separated variables for the classical hamiltonian systems governed by three classes of non-skew-symmetric, non-dynamical $gl\left(2\right)\otimes gl\left(2\right)$-valued classical r-matrices of the rational and trigonometric type. We show that for almost all r-matrices in the considered families the corresponding curves of separation are shifted spectral curves of the initial Lax matrices. The proposed scheme is illustrated by the examples of separation of variables for the integrable cases of the Kirckhoff problem based on the Lie algebra $gl\left(2\right)$ and on the considered families of the classical r-matrices.