Incarnations of XXX ̂𝔰𝔩_{𝔑} Bethe ansatz equations and integrable hierarchies

Abstract

We consider the space of solutions of the Bethe ansatz equations of the s l N ^ \widehat {\frak {sl}_N} XXX quantum integrable model, associated with the trivial representation of s l N ^ \widehat {\frak {sl}_N} . We construct a family of commuting flows on this space and identify the flows with the flows of coherent rational Ruijesenaars-Schneider systems. For that we develop in full generality the spectral transform for the rational Ruijesenaars-Schneider system.