The article is devoted to the study of R-matrix-valued Lax pairs for N -body (elliptic) Calogero-Moser models. Their matrix elements are given by quantum GLÑ R-matrices of Baxter-Belavin type. ForÑ = 1 the widely known Krichever's Lax pair with spectral parameter is reproduced. First, we construct the R-matrix-valued Lax pairs for Calogero-Moser models associated with classical root systems. For this purpose we study generalizations of the D'Hoker-Phong Lax pairs. It appeared that in the R-matrix-valued case the Lax pairs exist in special cases only. The number of quantum spaces (on which R-matrices act) and their dimension depend on the values of coupling constants. Some of the obtained classical Lax pairs admit straightforward extension to the quantum case. In the end we describe a relationship of the R-matrix-valued Lax pairs to Hitchin systems defined on SL NÑ bundles with nontrivial characteristic classes over elliptic curve. We show that the classical analogue of the anisotropic spin exchange operator entering the R-matrix-valued Lax equations is reproduced in these models.

Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the L-matrix.

We describe the correspondence of the Matsuo-Cherednik type between the quantum n-body Ruijsenaars-Schneider model and the quantum Knizhnik-Zamolodchikov equations related to supergroup GL(N |M ). The spectrum of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the Z 2 -grading for a fixed value of N + M , so that N + M + 1 different qKZ systems of equations lead to the same n-body quantum problem. The obtained results can be viewed as a quantization of the previously described quantumclassical correspondence between the classical n-body Ruijsenaars-Schneider model and the supersymmetric GL(N |M ) quantum spin chains on n sites.where the number of indices j k such that j k = a is equal to M a for all a = 1, . . . , K. The dual vectors J are defined in so that J J ′ = δ J,J ′ .Then the statement of the qKZ-Ruijsenaars correspondence is as follows [19]. For any solution of the qKZ equations (1.1) Φ = J Φ J J from the weight subspace V({M a }) the function Ψ = J Φ J , Φ J = Φ J (x 1 , ..., x n ) (1.7) 4 The quantum R-matrices entering (1.2) are assumed to be unitary: R ij (x)R ji (−x) = id. 5 The set {e ab | a, b = 1...K} is the standard basis in Mat(K, C): (e ab ) ij = δ ia δ jb .