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

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