Extra integrals of motion and the Lax representation are found for interacting spin systems with the Hamiltonian H= (J/2)Z~'k-l,j,~k ~(j-k)%ak, where one of the periods of the Weierstrass ~ function is equal to N. The Heisenberg and Haldane-Shastry chains appear as limiting cases of these systems at some values of the second period. The simplest eigenvectors and eigenvalues of H corresponding to the scattering of two spin waves are presented explicitly for these finite-dimensional systems and for their infinite-dimensional version.

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Starting from a Calogero-Sutherland model with hyperbolic interaction confined by an external field with Morse potential we construct a Heisenberg spin chain with exchange interaction ∝ 1/ sinh 2 x on a lattice given in terms of the zeroes of Laguerre polynomials. Varying the strength of the Morse potential the Haldane-Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with nonuniform nearest neighbour coupling in a nonuniform magnetic field which allows to study the thermodynamics in the limit of infinite chains.

For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a "site", we associate a vector space V µ 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 ∆), and spin exchange operators {P ρ } (ρ ∈ ∆) which exchange the spins at the sites µ and s ρ (µ). Here s ρ is the reflection generated by ρ. For each ∆ 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 CalogeroMoser model. For ∆ = A r and R = vector representation 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.