A family of commuting flows on the space of solutions of the Bethe ansatz equations of the $\widehat{\frak{sl}_N}$ XXX quantum integrable model, associated with the trivial representation of $\widehat{\frak{sl}_N}$ is constructed. The  family  is identified  with the flows of $N$-tuple of coherent rational  Ruijesenaars-Schneider systems. For that we develop in full generality the spectral transform for the rational  Ruijesenaars-Schneider system.