Formation of singularities of 2D soliton equations represented by $L,A,B$-triples

We expose recent results on a certain geometrical mechanism of the formation of singularities of modified Novikov-Veselov and Davey-Stewartson II equations. The singular solutions are constructed by means of surface theory. These systems are represented by L,A,B-triples and we discuss
the relation of such a mechanism to the degeneration of the zero level discrete spectra of the corresponding L-operators.