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The Wahlquist-Estabrook prolongation method allows to obtain for some PDEs aLie algebra that is responsible for Lax pairs and Backlund transformations ofcertain type. We study the Wahlquist-Estabrook algebra of the n-dimensionalgeneralization of the Landau-Lifshitz equation and construct an epimorphismfrom this algebra onto an infinite-dimensional quasigraded Lie algebra L(n) ofcertain matrix-valued functions on an algebraic curve of genus 1+(n-3)2^{n-2}.For n=3,4,5 we prove that the Wahlquist-Estabrook algebra is isomorphic to thedirect sum of L(n) and a 2-dimensional abelian Lie algebra. Using theseresults, for any n a new family of Miura type transformations (differentialsubstitutions) parametrized by points of the above mentioned curve isconstructed. As a by-product, we obtain a representation of L(n) in terms of afinite number of generators and relations, which may be of independentinterest.

Generalized Landau-Lifshitz systems and Lie algebras associated with higher genus curves