  We construct the classical r-matrix structure for the Lax formulation of BC_N
Ruijsenaars-Schneider systems proposed in hep-th 0006004. The r-matrix
structure takes a quadratic form similar to the A_N Ruijsenaars-Schneider
Poisson bracket behavior, although the dynamical dependence is more
complicated. Commuting Hamiltonians stemming from the BC_N
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.
