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 (C-Toda). It can be regarded as certain subhierarchy the 2D lattice obtained by imposing constraint $\bar {\cal L}={\cal L}^{\dag}$ on two Lax operators (in symmetric gauge). prove existence tau-function C-Toda and show that it is square root tau-function. In this some other respects analogue CKP hierarchy. also shown zeros elliptic solutions satisfy dynamical Ruijsenaars-Schneider model restricted to turning points in phase space. The spectral curve has holomorphic involution interchange marked Baker-Akhiezer function essential singularities.