Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-operators as the Calogero-Moser ones?

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.Comment: LaTeX, 18pp, a revised versio

The Classical rr-Matrix for the Relativistic Ruijsenaars-Schneider System

We compute the classical $r$-matrix for the relativistic generalization of the Calogero-Moser model, or Ruijsenaars-Schneider model, at all values of the speed-of-light parameter $\lambda$. We connect it with the non-relativistic Calogero-Moser $r$-matrix $(\lambda \rightarrow -1)$ and the $\lambda = 1$ sine-Gordon soliton limit.Comment: LaTeX file, no figures, 8 page

Linear rr-Matrix Algebra for Systems Separable\\ in Parabolic Coordinates

We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of $2\times 2$ matrices for the whole hierarchy, we construct the associated linear $r$-matrix algebra with the $r$-matrix dependent on the dynamical variables. A dynamical Yang-Baxter equation is discussed.Comment: 10 pages, LaTeX. Submitted to Phys.Lett.