  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.
