The Moutard transformation and its applications
We discuss the Moutard transformation which is a two-dimensional generalization of the Darboux transformation of the Schroedinger operator and expose some applications of this transformations to spectral theory and soliton equations. In particular, we expose examples of Schroedinger operators with smooth fast-decaying potentials and nontrivial kernels in $L_2$ and of blowing up solutions of the Novikov--Veselov equation, a two-dimensional generalization of the Korteweg--de Vries equation. These results we obtained jointly with S.P. Tsarev.