We construct the classical r-matrix structure for the Lax formulation of BCN Ruijsenaars–Schneider systems proposed in Commun. Math. Phys., 115, 127 (1988). The r-matrix structure takes a quadratic form similar to the AN Ruijsenaars–Schneider Poisson bracket behavior, although the dynamical dependence is more complicated. Commuting Hamiltonians stemming from the BCN Ruijsenaars–Schneider Lax matrix are shown to be linear combinations of particular Koornwinder–van Diejen “external fields” Ruijsenaars–Schneider models, for specific values of the exponential one-body couplings. Uniqueness of such commuting Hamiltonians is established once the first of them and the general analytic structure are given.

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