  The Wahlquist-Estabrook prolongation method allows to obtain for some PDEs a
Lie algebra that is responsible for Lax pairs and Backlund transformations of
certain type. We study the Wahlquist-Estabrook algebra of the n-dimensional
generalization of the Landau-Lifshitz equation and construct an epimorphism
from this algebra onto an infinite-dimensional quasigraded Lie algebra L(n) of
certain 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 the
direct sum of L(n) and a 2-dimensional abelian Lie algebra. Using these
results, for any n a new family of Miura type transformations (differential
substitutions) parametrized by points of the above mentioned curve is
constructed. As a by-product, we obtain a representation of L(n) in terms of a
finite number of generators and relations, which may be of independent
interest.
