^{\mu} \big| _{z= a^{-}} ^{z = a^{+}} = 0 \;\;\;\;\ , \;\;\;\;\ \left(
\partial _{z} A ^{\mu} \right) \big| _{z= a^{-}} ^{z = a^{+}} = - \tilde{\theta}
\epsilon ^{3 \mu \alpha} _{\;\;\;\;\ \nu} \partial _{\alpha} A ^{\nu } \big| %
_{z=a} .  \label{BC4Pot}$$ One can further check that these boundary conditions for the $4$-potential corresponds to the ones obtained in Eqs. (\[Ampere-BC\]-\[Faraday-BC\]).

To obtain a general solution for the potentials $\phi$ and $\mathbf{A}$ in the presence of arbitrary external sources $j^\mu (\textbf{x})$, we introduce the GF $G ^{\nu} _{\; \sigma} \left( \mathbf{x} , 
\mathbf{x} ^{\prime} \right)$ solving Eq. (\[FiedlEqPlaneConfig\]) for a point-like source, $$\left[ - \eta ^{\mu} _{\; \nu} \nabla ^{2} - \tilde{\theta}\delta \left(
z-a\right) \epsilon _{\;\;\;\;\ \nu }^{3 \mu \alpha }\partial _{\alpha }%
\right] G ^{\nu} _{\; \sigma} \left( \mathbf{x} , \mathbf{x} ^{\prime}
\right) = 4 \pi \eta ^{\mu} _{\; \sigma} \delta ^{3} \left( \mathbf{x} - 
\mathbf{x} ^{\prime} \right) ,  \label{EqsGreenMatrix}$$ together with the boundary conditions (\[BC4Pot\]), in such a way that the general solution for the $4$-potential in the Coulomb gauge is $$A ^{\mu} \left( \mathbf{x} \right) = \int d ^{3} \mathbf{x} ^{\prime} \; G
^{\mu} _{\; \nu} \left( \mathbf{x} , \mathbf{x} ^{\prime} \right) j^{\nu}
\left( \mathbf{x} ^{\prime} 