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We develop a general scheme to construct integrable systems starting fromrealizations in symmetric coboundary dynamical Lie algebroids and symmetriccoboundary Poisson groupoids. The method is based on the successive use ofDirac reduction and Poisson reduction. Then we show that certain spinCalogero-Moser systems associated with symmetric Lie subalgebras can be studiedin this fashion. We also consider some spin-generalized Ruisjenaars- Schneiderequations which correspond to the $N$-soliton solutions of $A^{(1)}_n$ affineToda field theory. In this case, we show how the equations are obtained fromthe Dirac reduction of some Hamiltonian system on a symmetric coboundarydynamical Poisson groupoid.Comment: 48 page

Poisson Involutions, Spin Calogero-Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models