On the (Non)Removability of Spectral Parameters in Z 2 $\mathbb{Z}_{2}$ -Graded Zero-Curvature Representations and Its Applications

Abstract

We generalise to the $\mathbb{Z}_2$-graded set-up a practical method for inspecting the (non)removability of parameters in zero-curvature representations for partial differential equations (PDEs) under the action of smooth families of gauge transformations. We illustrate the generation and elimination of parameters in the flat structures over $\mathbb{Z}_2$-graded PDEs by analysing the link between deformation of zero-curvature representations via infinitesimal gauge transformations and, on the other hand, propagation of linear coverings over PDEs using the Fr\"olicher--Nijenhuis bracket.