  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 ${\rm GL}_{\tilde N}$ $R$-matrices of Baxter-Belavin type. For $\tilde
N=1$ 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
${\rm SL}_{N\tilde N}$ 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.
