This question illustrates a more general issue that often arises with partial differential equations. There are similar issues in General Relativity for example.
The first thing to note is that a given source distribution is almost never enough to settle uniquely a solution to a set of partial differential equations, because you can add in whatever functions satisfy the source-free equations and you still have a solution. Physically, if you have a point charge then around it you may find a Coulomb-law field, or you may find a Coulomb-law field plus electromagnetic waves. (For linear equations this 'adding in' is exactly an addition operation; for non-linear equations it is not simply addition but a similar conclusion holds.)
It follows that to handle this issue in general you need either a Cauchy surface on which the fields have been given, or else further conditions in addition to the source distribution. Symmetry is often brought in to play that role.
