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Our GBDT version of B\"acklund-Darboux transformation is applied to theconstruction of wide classes of new explicit solutions of self-adjoint andskew-self-adjoint Dirac systems, dynamical Dirac and Dirac--Weyl systems. Thatis, we construct explicit solutions of systems with non-vanishing at infinitypotentials. In particular, the cases of steplike potentials and power growth ofpotentials are treated. It is essential (especially, for dynamical case) thatthe generalised matrix eigenvalues are used in GBDT instead of the usualeigenvalues (and those matrix eigenvalues are not necessarily diagonal). Theconnection of Dirac--Weyl system with graphene theory is discussed. Explicitexpressions for Weyl--Titchmarsh functions are derived.

On new classes of explicit solutions of Dirac, dynamical Dirac and  Dirac--Weyl systems with non-vanishing at infinity potentials, their  properties and applications