Electromagnetism uniqueness theorem

The electromagnetism uniqueness theorem states that providing boundary conditions for Maxwell's equations uniquely fixes a solution for those equations.
However, this theorem must not be misunderstood as that providing boundary conditions (or the field solution itself) uniquely fixes a source distribution. One counterexample is that the field outside a uniformly charged sphere may also be produced by a point charge placed at the center of the sphere instead, i.e. the source needed to produce such field at a boundary outside the sphere is not unique.