On ${\cal N}{=}\,2$ supersymmetric Ruijsenaars-Schneider models

We construct an ${\cal N}{=}\,2$ supersymmetric extension of $n$-particle Ruijsenaars-Schneider models. The guiding feature is a deformation of the phase space. The supercharges have a "free" form linear in the fermions but produce an interacting four-fermion Hamiltonian. A field-dependent unitary transformation maps to standard fermions obeying conventional Poisson brackets. In this frame, the supercharges and Hamiltonian have long "fermionic tails". We also comment on previous attempts in this direction.