  The inverse scattering method for the Novikov-Veselov equation is studied for
a larger class of Schr\"odinger potentials than could be handled previously.
Previous work concerns so-called conductivity type potentials, which have a
bounded positive solution at zero energy and are a nowhere dense set of
potentials. We relax this assumption to include logarithmically growing
positive solutions at zero energy. These potentials are stable under
perturbations. Assuming only that the potential is subcritical and has two weak
derivatives in a weighted Sobolev space, we prove that the associated
scattering transform can be inverted, and the original potential is recovered
from the scattering data.
