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The Moutard transformation for a two-dimensional Dirac operator with acomplex-valued potential is constructed. It is showed that this transformationrelates the potentials of Weierstrass representations of surfaces related by acomposition of the inversion and a reflection with respect to an axis. It isgiven an analytical description of an explicit example of such a transformationwhich results in a creation of double points on the spectral curve of a Diracoperator with a double-periodic potential.

The Moutard transformation of two-dimensional Dirac operators and conformal geometry of surfaces in the four-space