In 1988 P.G. Grinevich and S.P. Novikov showed that in the fixed-energy scattering problem for the two-dimensional Schrodinger operator one has to study Vekua-Bers generalized analytic functions with special contour singularities. For sufficiently long period no approaches for studying such problems were known. Recently we understood that the Moutard transformation can be applied to the study of generalized analytic functions with such special singularities