Over on Physics.SE there is an interesting question about electrostatic configurations where all electric field lines are straight. Clearly, setups with spherical, cylindrical, or planar symmetry are examples, but we want to know if there are more possibilities. By considering the equipotential surfaces, we've managed to reduce this problem to a pure math question.
Specifically, the first equipotential surface needs to have constant mean curvature. To construct the next equipotential surface, we take every point on this first surface, and follow the normal vector outward a uniform distance, and so on. The electric field lines will be straight precisely when all the surfaces constructed this way have constant mean curvature. (For example, if you start with an infinite cylinder, you get a family of concentric infinite cylinders.)
Are there examples that satisfy this criterion that don't have spherical, cylindrical, or planar symmetry?