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 (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint ℒ̅=ℒ^† on the two Lax operators (in the symmetric gauge). We prove the existence of the tau function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau function. In this and some other respects, the C-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 interchanges the marked points in which the Baker–Akhiezer function has essential singularities.