The second order pullback equation

Let f, g be two closed k-forms over R-n. The pullback equation studies the existence of a diffeomorphism phi : R-n -> R-n such that phi*(g) = f. We prove two types of results. The first one sharpens some of the existing regularity results. The second one discusses the possibility of choosing the map phi as the gradient of a function Phi : R-n -> R. We show that this is a very rare event unless the two forms are constant.