Trigonometric real form of the spin RS model of Krichever and Zabrodin

In 1995, Krichever and Zabrodin introduced interesting spin extensions of the Ruijsenaars–Schneider system, working at the level of equations of motion. Their investigation and all earlier studies of the Hamiltonian interpretation of the model used complex holomorphic settings. Based on a joint paper with Fairon and Marshall (arXiv:2007.08388), we explain that the trigonometric real form of the model emerges from Hamiltonian reduction of a ‘free particle’ supported by a spin extension of the Heisenberg double of the U(n) Poisson–Lie group. Then, we characterize the Hamiltonian structure of the real trigonometric spin RS model and demonstrate its degenerate integrability.