  We demonstrate that in a certain gauge the Lax matrices of the rational and
hyperbolic Ruijsenaars--Schneider models have a quadratic $r$-matrix Poisson
bracket which is an exact quadratization of the linear $r$--matrix Poisson
bracket of the Calogero--Moser models. This phenomenon is explained by a
geometric derivation of Lax equations for arbitrary flows of both hierarchies,
which turn out to be governed by the same dynamical $R$--operator.
