Time discretization of the deformed Ruijsenaars-Schneider system

We obtain B\"acklund transformations and integrable time discretization of 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. We also show that the deformed Ruijsenaars-Schneider system in discrete time is the dynamical system for poles of elliptic solutions to the fully discrete Kadomtsev-Petviashvili equation of type B. Besides, we suggest a field analogue of the deformed Ruijsenaars-Schneider system on a space-time lattice.
Comment: 18 pages, no figures