Universal Lax pairs for spin Calogero-Moser models and spin exchange models

For any root system Delta and a set of vectors R which form a single orbit of the reflection (Weyl) group G(Delta) generated by Delta, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member mu of R, to be called a 'site', we associate a vector space V-mu whose element is called a 'spin'. Its dynamical variables are the canonical coordinates [q(j), p(j)} of a particle in R-r (r = rank of Delta) and spin exchange operators {(P) over cap (rho)} (rho is an element of Delta) which exchange the spins at the sites mu and s(rho)(mu). Here s(rho) is the reflection generated by rho. For each Delta and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by 'freezing' the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For Delta = A(r) and R = set of vector weights it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.