Loop algebras and symmetries related to both a compact and a noncompact version of a (2+1)-dimensional spin model

Abstract

We show that both the compact and the noncompact version of a nonlinear spin field system in 2 + 1 dimensions, the so-called Ishimori model, allows an infinite-dimensional Lie algebra with a loop algebra structure. This feature seems characteristic of any integrable nonlinear field equation in 2 + 1 dimensions. We find also that the corresponding group transformations lead to a reduction procedure from which new solutions can be obtained from known ones.