I am trying to find either
- a general solution
- a non-trivial class of solutions
to the following equation:
\begin{equation} (D_i D_j + D_j D_i) 1 = s_{ij} \end{equation}
where $D_i = \partial_i - a_i$ is a derivative (I think), $s_{ij}$ is an arbitrary symmetric tensor, and 1 is just the real number. (By solution, I mean I am ideally looking for $a$ in terms of $s$.) If it helps, the answer can be valid only in 3 dimensions.