fields from the modified Maxwell equations together with the boundary conditions (\[Ampere-BC\]-\[Faraday-BC\]), however, just as in ordinary electrodynamics, there might be occasions where information about the sources is unknown and rather we are provided with information of the 4-potential at the given boundaries. In these cases, the GF method provides the general solution, to a given boundary-value problem (Dirichlet or Neumann) for arbitrary sources. Nevertheless, in the sequel we restrict ourselves to contributions of free sources outside the $\theta$-boundary, and without boundary conditions imposed on additional surfaces, except for the standard boundary conditions at infinity.

Since the homogeneous Maxwell equations that express the relationship between potentials and fields are not modified, the electrostatic and magnetostatic fields can be written in terms of the $4$-potential $A ^{\mu} = \left( \phi , 
\mathbf{A} \right)$ according to $\mathbf{E} = - \nabla \phi $ and $\mathbf{B%
} = \nabla \times \mathbf{A}$ as usual. In the Coulomb gauge $\nabla \cdot 
\mathbf{A} = 0 $, the $4$-potential satisfies the equation of motion $$\left[ - \eta ^{\mu} _{\; \nu} \nabla ^{2} - \tilde{\theta}\delta \left(
z-a\right) \epsilon _{\;\;\;\;\ \nu }^{3 \mu \alpha }\partial _{\alpha }%
\right] A^{\nu }=4 \pi j^{\mu },  \label{FiedlEqPlaneConfig}$$ together with the boundary conditions $$A 