A nonlinear scattering transform is studied for the two-dimensional Schrodinger equation at zero energy with a radial potential. For a class of potentials call "conductivity type", it is known that there are no singularities in this scattering transform. We will look at a family of perturbations of conductivity type potentials and show where the singularities in their scattering transforms occur. This is some of the first work explicitly calculating the behavior of these singularities.