On integrability of the deformed Ruijs']jsenaars-Schneider system

We find integrals of motion for the recently introduced deformed Ruijsenaars-Schneider many-body system, which is the dynamical system for poles of elliptic solutions to the Toda lattice with constraint of type B. Our method is based on the fact that the equations of motion for this system coincide with those for pairs of Ruijsenaars-Schneider particles which stick together preserving a special fixed distance between the particles. We also obtain B<spacing diaeresis>acklund transformations and integrable time discretization of the deformed Ruijsenaars-Schneider system, which is shown to be the dynamical system for poles of elliptic solutions to the fully discrete Kadomtsev-Petviashvili equation of type B. In additon, we propose a field analogue of the deformed Ruijsenaars-Schneider system on a space-time lattice.