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

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