Nondegenerate curves onS 2 and orbit classification of the Zamolodchikov algebra

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The Zamolodchikov algebra is the next case after the Virasoro algebra in the natural hierarchy of the Poisson structures on linear differential equations. We describe here the complete classification of the symplectic leaves of this algebra. It turns out that each symplectic leaf is uniquely defined by the conjugacy class of the monodromy operator and one discrete (2- and 3-valued) invariant arising from the homotopy classes of nondegenerate curves.