equations [@26]. Other authors state that it is an *integrability condition* of these equations [@27; @28]. Some other authors are more cautious and claim that Maxwell’s equations are *consistent* with the continuity equation [@23; @29]. Although Maxwell’s equations formally imply the continuity equation, the idea that the latter is a consequence of the former is in a sense questionable. The fact is that the continuity equation has its own existence *independent* of Maxwell’s equations. This can be illustrated by the fact that there are field equations of different electromagnetic theories that are also consistent with the continuity equation. For example, one of these theories arises when the Faraday induction term of Maxwell’s equations is eliminated, obtaining the field equations of a Galilean-invariant instantaneous electrodynamics [@30; @31]. Other examples are the Proca equations of the massive electrodynamics [@32] and the field equations of an electrodynamics in an Euclidean four-space [@33; @34]. Therefore, one should interpret the continuity equation as a formal representation of the principle of charge conservation, but having always in mind that this principle is not exclusive of Maxwell’s theory.

Accordingly, we can equally use the continuity equation to formulate other existence theorems for potentials or fields which can 