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We consider the Moutard transformation which is a two-dimensional version ofthe well-known Darboux transformation. We give an algebraic interpretation ofthe Moutard transformation as a conjugation in an appropriate ring and thecorresponding version of the algebro-geometric formalism for two-dimensionalSchroedinger operators. An application to some problems of the spectral theoryof two-dimensional Schroedinger operators and to the $(2+1)$-dimensionalNovikov--Veselov equation is sketched.

The Moutard transformation: an algebraic formalism via pseudodifferential operators and applications