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Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations

We give detailed description for continuous version of the classical IRF-Vertex relation, where on the IRF side we deal with the Calogero-Moser-Sutherland models. Our study is based on constructing modifications of the Higgs bundles of infinite rank over elliptic curve and its degenerations. In this way the previously predicted gauge equivalence between L-A pairs of the Landau-Lifshitz type equations and 1+1 field theory generalization of the Calogero-Moser-Sutherland models is described. In this paper the ${\rm sl}_2$ case is studied. Explicit changes of variables are obtained between the rational, trigonometric and elliptic models.