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 t1 move as particles of the elliptic RuijsenaarsSchneider 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 mth hierarchical times t(m) and (t) over bar (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.