  A nonlinear scattering transform is studied for the two-dimensional
Schrodinger equation at zero energy with a radial potential. First explicit
examples are presented, both theoretically and computationally, of potentials
with nontrivial singularities in the scattering transform. The singularities
arise from non-uniqueness of the complex geometric optics solutions that define
the scattering transform. The values of the complex spectral parameter at which
the singularities appear are called exceptional points. The singularity
formation is closely related to the fact that potentials of conductivity type
are critical in the sense of Murata.
