Structure of prescribed gradient domains for non-integrable vector fields

Let $F in C^1(Omega, ^{n})$ and $fin C^2(Omega)$, where $Omega$ is an open subset of $ ^{n}$ with $n$ even. We describe the structure of the set of points in $Omega$ at which the equality $D f = F$ and a certain non-integrability condition on $F$ hold. This result generalizes the second statement of cite[Theorem 3.1]{Ba}.