The Moutard Transformation of Two-Dimensional Dirac Operators and the Conformal Geometry of Surfaces in Four-Dimensional Space

The Moutard transformation for the two-dimensional Dirac operator with complex-valued potential is constructed. It is shown that this transformation binds the potentials of Weierstrass representations of the surfaces related by the composition of inversion and reflection with respect to the axis. An explicit analytic example of a transformation leading to the appearance of double points on the spectral curve of the Dirac operator is described analytically.