Separation of variables for linear Lax algebras and classical r-matrices

We consider the problem of separation of variables for the algebraically integrable Hamiltonian systems possessing gl(n)-valued Lax matrices depending on a spectral parameter that satisfy linear Poisson brackets with some gl(n)⊗gl(n)-valued classical r-matrices. We formulate, in terms of the corresponding r-matrices, a sufficient condition that guarantees that the “separating polynomials” of E.Sklyanin, Comm. Math. Phys. 150, 181 (1992), D.Scott, J. Math. Phys. 35, 5831 (1994), M.Gekhtman, Comm. Math. Phys. 167, 593 (1995), P.Diener, B.Dubrovin, Algebraic-geometrical Darboux coordinates in R-matrix formalism, SISSA preprint 88-94-FM (1994), produce a system of canonical variables. We consider two examples of classical r-matrices and separating polynomials. One of these examples, namely, the n-parametric family of non-skew-symmetric non-dynamical classical r-matrices of T.Skrypnyk, Phys. Lett. A 334, 390, and 347, 266 (2005) and the corre- sponding separating polynomials is new. We show that the separating polynomials of P.Diener, B.Dubrovin, ibid., produce in this case a complete set of separated variables for the corresponding generalized Gaudin models with or without external magnetic field