Nontrivial 1-parameter families of zero-curvature representations obtained via symmetry actions

Abstract

In this paper we consider the problem of constructing a 1 -parameter family α λ of zero-curvature representations of an equation E , by means of classical symmetry actions on a given zero-curvature representation α . By using the cohomology defined by the horizontal gauge differential of α , we provide an infinitesimal criterion which permits to identify all infinitesimal classical symmetries of E whose flow could be used to embed α into a nontrivial 1 -parameter family α λ of zero-curvature representations of E . The results of the paper are illustrated with some examples.