  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.
