Einstein, $\sigma$-model and Ernst-type equations and non-isospectral GBDT version of Darboux transformation

Abstract

We present a non-isospectral GBDT version of B\"acklund-Darboux transformation for the gravitational and $\sigma$-model equations. New families of explicit solutions correspond to the case of GBDT with non-diagonal generalized matrix eigenvalues. An interesting integrable Ernst-type system, the auxiliary linear systems of which are non-isospectral canonical systems, is studied as well.