ct 2 00 2 Prolongation structure of the Krichever - Novikov equation

We completely describe Wahlquist-Estabrook prolongation structures (coverings) dependent on u, ux, uxx, uxxx for the Krichever-Novikov equation ut = uxxx−3uxx/(2ux)+p(u)/ux+aux in the case when the polynomial p(u) = 4u − g2u − g3 has distinct roots. We prove that there is a universal prolongation algebra isomorphic to the direct sum of a commutative 2-dimensional algebra and a certain subalgebra of the tensor product of sl2(C) with the algebra of regular functions on an affine elliptic curve. This is achieved by identifying this prolongation algebra with the one for the anisotropic Landau-Lifshitz equation. Using these results, we find for the Krichever-Novikov equation a new zero-curvature representation, which is polynomial in the spectral parameter in contrast to the known elliptic ones. Mathematics Subject Classification (2000): 37K10, 37K30, 35Q53