Finding Solutions to a PDE System with Known Scalar Function

[vibe3]

Hi all, I am looking for ways to solve the following system of equations for $\vec{B}$:
$$\vec{B} \cdot \nabla f = 0$$
$$\left( \nabla \times \vec{B} \right) \cdot \nabla f = 0$$
$$\nabla \cdot \vec{B} = 0$$
and $f$ is a known scalar function. I think we can assume there is a solution since we have 3 equations and 3 unknown components of $\vec{B}$.

I don't think there is an analytic solution here, but could someone give some pointers on how one would solve this system numerically?

 

[hunt_mat]

Method of characteristics I think?