electromagnetism - Unique eigenvectors of the EM energy-momentum tensor

This is a two-part question: 1) I understand that every tensor in a Minkowski 4-space has four eigenvectors, of which only one is timelike. Under what circumstances, if any, does an EM energy-momentum tensor have four unique unit eigenvectors? 2) Is there a formula for the eigenvectors of the EM energy-momentum tensor in terms of $A^\mu$, $F^{\mu\nu}$ or $\vec E$ and $\vec B$?