So let's assume we know that $\vec{\nabla}\times\vec{v}=5\hat{\imath} + 3\hat{k}$ and $\vec{\nabla}\cdot{\vec{v}}=0$ ($\hat{\imath}$, $\hat{\jmath}$, $\hat{k}$ being the unit vectors in the $x$, $y$ and $z$ direction). What whould be a general/good/efficient/clever process of finding $\vec{v}$?

Edit: Trying to clarify things a bit: the reason for my question is the fact, that Maxwell's equations define the curl of the magnetic and the electric field. So I'm wondering how to find a vector if all I know is it's curl and it's div (I assume it's constant component to be zero).

In contrast to that, finding $\vec{\nabla}\times{\vec{v}}$ and $\vec{\nabla}\cdot{\vec{v}}$ is simple if $\vec{v}$ is known. But what kind of approach is advisable if I want to go the oppesit direction?