Separation of variables, Lax-integrable systems and [formula omitted]-valued classical [formula omitted]-matrices

Abstract

In the present paper we consider a problem of separation of variables for Lax-integrable hamiltonian system generated by general non-skew-symmetric gl(2)⊗gl(2)-valued classical r-matrix r(u,v) with spectral parameters. We specify the general separability condition of Dubrovin and Skrypnyk (2018) on the components of the r-matrix and consider two new classes of examples of classical non-skew-symmetric r-matrices r(u,v) for which the separability condition is satisfied and the whole construction produces a complete set of canonical coordinates. For each of these r-matrices we consider in details the corresponding classical Gaudin-type models with N spins. In the case N=2 we explicitly find coordinates and momenta of separation, reconstruction formulae and Abel–Jacobi quadratures. Some potential physical applications of the considered models are discussed.