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

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


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Luiz Alberto de Oliveira Silva