The following seemingly-simple problem came up when working on a problem in the fluid theory of plasmas.
Given a vector field $\mathbf{A}$, find a symmetric tensor $\mathbf{P}$ such that $\boldsymbol{\nabla}\times\mathbf{A} = \boldsymbol{\nabla}\cdot\mathbf{P}$.
This isn't very hard if you don't require $\mathbf{P}$ to be symmetric (it's just $P_{ij} = \epsilon_{ijk}A_k$), but the symmetry requirement is throwing me off.