Constrained Toda hierarchy and turning points of the Ruijsenaars-Schneider model

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy. It can be regarded as a subhierarchy of the 2D Toda lattice obtained by imposing a certain constraint connecting the two Lax operators of the latter. We prove the existence of the tau-function of the constrained Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the constrained Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities. This is a joint work with I.Krichever.