Why is the Ruijsenaars-Schneider hierarchy governed by the same R-operator as the Calogero-Moser one?

We demonstrate that in a certain gauge the Lax matrix of the Ruijsenaars-Schneider (RS) model has a quadratic r -matrix structure independent of the relativistic parameter of the model. In particular, the dynamical r -matrix object governing the hierarchy of Lax equations connected with the RS model, turns out to be identical with the r -matrix of the Calogero-Moser (CM) model. In the degenerate cases (hyperbolic and rational) this phenomenon is partly explained by a geometric derivation of Lax equations for arbitrary flows of both the RS and the CM hierarchies.