Nonlocal pseudosymmetries and Bäcklund transformations as C -morphisms

In this paper, we show how factorisation with respect to nonlocal pseudosymmetries enables us to obtain Bäcklund transformations, which are described as nonlocal C -morphisms of differential equations. It follows that, in this approach, Bäcklund transformations are determined by basic invariants of the exploited nonlocal pseudosymmetries. As illustrated in several representative examples, including one involving a new integrable equation, our resulting framework enables a very general structural approach, distinct from the case-by-case ones often adopted in the literature—especially in the analysis of cases where the sought Bäcklund transformations also affect the independent variables.