Given a two-dimensional vector field $U(x,y) \in C^2(\mathbb{R}^2, \mathbb{R}^2)$, consider the expression \begin{align} \operatorname{grad} \operatorname{div} U(x,y) = V(x,y) \end{align} for some output vector field $V(x,y) \in C^0(\mathbb{R}^2, \mathbb{R}^2)$.
Question: Given a vector field $V(x,y)$, is there a straightforward way to find a vector field $U(x,y)$ such that $\operatorname{grad} \operatorname{div} U(x,y) = V(x,y)$?
Note that constant vector fields live in the nullspace of $\operatorname{grad} \operatorname{div}(\cdot)$, so such $U(x,y)$ cannot be unique! I have thought about somehow using the Helmholtz decomposition, but it is unclear to me how to proceed.