Lax Triples for Integrable Surfaces in Three-Dimensional Space

Abstract

We study Lax triples (i.e., Lax representations consisting of three linear equations) associated with families of surfaces immersed in three-dimensional Euclidean spaceE3. We begin with a natural integrable deformation of the principal chiral model. Then, we show that all deformations linear in the spectral parameterλare trivial unless we admit Lax representations in a larger space. We present an explicit example of triply orthogonal systems with Lax representation in the groupSpin(6). Finally, the obtained results are interpreted in the context of the soliton surfaces approach.