If I have an electric field that its direction is parallel to the direction of the wave propagation, it will not satisfy Gauss's law for vacuum. However we can say it satisfies Gauss's law for non-zero charge density (mathematically). For this case:
Does that mean this electric field is practically feasible for an EM wave (for non-zero charge densities)?
Will the other 3 laws of Maxwell be violated?
Furthermore, If the direction of the wave propagation is perpendicular to the direction of the electric field, does this guarantee that the field vector satisfy all Maxwell's equation?