Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model

We consider solutions of the 2D Toda lattice hierarchy that are elliptic functions of the “zeroth” time $t_0=x$. It is known that their poles as functions of $t_1$ move as particles of the elliptic Ruijsenaars–Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians that govern the dynamics of poles with respect to the $m$th hierarchical times $t_m$ and $\bar t_m$ of the 2D Toda lattice hierarchy are obtained from the expansion of the spectral curve for the Lax matrix of the Ruijsenaars–Schneider model at the marked points.