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We use the inverse scattering method to construct classical solutions for theNovikov-Veselov (NV) equation, solving a problem posed by Lassas, Mueller,Siltanen, and Stahel. We exploit Bogadanov's Miura-type map which transformssolutions of the modified Novikov-Veselov (mNV) equation into solutions of theNV equation. We show that the Cauchy data of conductivity type considered byLassas, Mueller, Siltanen, and Stahel correspond precisely to the range of theMiura map, so that it suffices to study the mNV equation. We solve the mNVequation using the scattering transform associated to the defocussingDavey-Stewartson II equation.

Miura Maps and Inverse Scattering for the Novikov-Veselov Equation