Two-dimensional Periodic Schr\"odinger Operators Integrable at Energy Eigenlevel

The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schr\"odinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Shr\"odinger equation at the zero energy level is a smooth $M$-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schr\"odinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov--Veselov construction.