Relativistic interacting integrable elliptic tops

TL;DR

This paper introduces a relativistic extension of integrable elliptic tops that generalizes known models, providing new Lax pairs and connecting to the Ruijsenaars-Schneider model.

Contribution

It presents a novel relativistic generalization of elliptic integrable systems for interacting ${ m gl}(N)$ tops, including explicit Lax pairs on elliptic curves.

Findings

Models reproduce the spin elliptic ${ m GL}(M)$ Ruijsenaars-Schneider system for N=1

In the M=1 case, models become relativistic ${ m GL}(N)$ elliptic tops

Lax pairs with spectral parameter on elliptic curve are constructed

Abstract

We propose relativistic generalization of integrable systems describing $M$ interacting elliptic ${\rm gl}(N)$ tops of the Euler-Arnold type. The obtained models are elliptic integrable systems, which reproduce the spin elliptic ${\rm GL}(M)$ Ruijsenaars-Schneider model for $N=1$ case, while in the $M=1$ case they turn into relativistic integrable ${\rm GL}(N)$ elliptic tops. The Lax pairs with spectral parameter on elliptic curve are constructed.