For certain initial data, we solve the Novikov-Veselov equation by the inverse scattering method. The Novikov-Veselov equation is a (2+1)-dimensional completely integrable system that generalizes the (1+1)-dimensional Korteweg-de-Vries equation. The scattering maps were studied for conductivity-type potentials by Nachman, He proves existence and uniqueness properties of the exponentially growing solutions for the stationary Schrodinger equation with conductivity-type potentials. Our work expands the set of potentials for which his analysis holds. We then use these results to prove that the inverse scattering method yields classical solutions to the Novikov-Veselov equation for our larger class of potentials