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In 1988 P.G. Grinevich and S.P. Novikov showed that the fixed-energy inverse scattering problem for the two-dimensional Schrodinger operator with large potentials requires the study of generalized analytic functions with very special contour singularities. We show that exactly these singularities can be naturally generated or removed by applying the Moutard-type transformations, therefore we obtain an efficient tool for studying large date inverse problems.