  We present a new class of electrostatics problems that are exactly solvable
by adding finitely many image charges. Given a charge at some location inside a
cavity bounded by up to four conducting grounded segments of spheres: if the
spheres have a symmetry derived via a stereographic projection from a 4D finite
reflection group, then this is a solvable generalization of the familiar
problem of a charge inside a spherical cavity. There are 19 three-parametric
families of finite groups formed by inversions relative to at most four
spheres, each member of each family giving a solvable problem. We solve a
sample problem which derives from the reflection group $\mathbf{D}_{4}$ and
requires 191 image charges.
