Lax equations for relativistic ${\rm GL}(NM,{\mathbb C})$ Gaudin models on elliptic curve

Abstract: We describe the most general ${\rm GL}{NM}$ classical elliptic finite-dimensional integrable system, which Lax matrix has $n$ simple poles on elliptic curve. For $M=1$ it reproduces the classical inhomogeneous spin chain, for $N=1$ it is the Gaudin type (multispin) extension of the spin Ruijsenaars-Schneider model, and for $n=1$ the model of $M$ interacting relativistic ${\rm GL}_N$ tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on ${\rm GL}$-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars-Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general ${\rm GL}{NM}$ model is also described through $R$-matrices satisfying associative Yang-Baxter equation. This description provides the trigonometric and rational analogues of ${\rm GL}_{NM}$ models.