Divergenceless Vector Fields

We now resume the development by recalling that the wave operator $\mathcal A$ satisfies

$$\left[ \partial_t~\partial_z~\partial_x~\partial_y \right] \mathcal A \left[ \begin{array}{c} \phi\\ \vec A \end{array} \right]=\vec 0$$ (664)

for all four-vectors $\left[ \begin{array}{c} \phi\\ \vec A \end{array} \right]$ , and that, as a consequence, the wave Eq.() implies that $\left[ \begin{array}{c} \rho\\ \vec J \end{array} \right]$ has zero divergence (which is an expression of charge conservation)

$$\left[ \partial_t~\partial_z~\partial_x~\partial_y \right] \left[ \begin{array}{c} \rho\\ J_z\\ J_x\\ J_y \end{array} \right]=0$$ (665)

whenever a solution exists. Our interest lies in the converse:
Given that the source $\left[ \begin{array}{c} \rho\\ \vec J \end{array} \right]$ satisfies charge conservation, does there exist a solution to the Maxwell wave equation?
An affirmative answer is obtained by construction. It is based on decomposing the source four-vector into a linear combination each part of which is so simple that it separately satisfies charge conservation. This decomposition is

$$\left[ \begin{array}{c} \rho\\ J_z\\ J_x\\ J_y \end{array} \right... ...array} \right]J}_{~~~~\equiv \vec {\mathcal W}^{(3)}I+\vec {\mathcal W}^{(4)}J}$$ (666)

Charge conservation, $\partial_t+\vec \nabla \cdot \vec J =0$ , holds for all scalar fields $S^{TE},S^{TM},I$ and $J$ provided the latter two satisfy

$$0=\left(-\partial_t^2+\partial_z^2\right)J+ \left(\partial_x^2+\partial_y^2\right)I~~ \left( =\partial_t+\vec \nabla \cdot \vec J \right)~.$$ (667)