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 Sklyanin [Commun. Math. Phys. 150, 181 (1992)], Scott [J. Math. Phys. 35, 5831 (1994)], Gekhtman [Commun. Math. Phys. 167, 593 (1995)], and Diener and Dubrovin (Algebraic-geometrical Darboux coordinates in R-matrix formalism, SISSA, Preprint Report No. 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 Skrypnyk [Phys. Lett. A 334, 390 (2005); 347, 266 (2005)] and the corresponding separating polynomials is new. We show that the separating polynomials of Diener and Dubrovin produce in this case a complete set of separated variables for the corresponding generalized Gaudin models with or without external magnetic field.
For technical reasons, we deal with the eigenvectors of the adjoint operator writing them as row-vectors, i.e., as elements of the dual space.
For the Hamiltonian systems admitting Lax representation with the spectral parameter, it will naturally coincide with the spectral parameter in separating polynomials.
Needless to say that the assumption holds true for a generic point in the phase space.