\right) .  \label{GreenMatrix}$$ According to Eqs. (\[EqsGreenMatrix\]) the diagonal entries of the GF matrix are related with the electric and magnetic fields arising from the charge and current density sources, respectively, although they acquire $\theta$-dependence. However, the non-diagonal terms encode the magnetoelectric effect, *i.e.* the charge (current) density contributing to the magnetic (electric) field.

In the following we concentrate in constructing the solution to Eq. (\[EqsGreenMatrix\]). The GF we consider has translational invariance in the directions parallel to $\Sigma $, that is in the transverse $x$ and $y$ directions, while this invariance is broken in the $z$ direction. Exploiting this symmetry we further introduce the Fourier transform in the direction parallel to the plane $\Sigma$, taking the coordinate dependence to be $\left( \mathbf{x}-\mathbf{x}^{\prime }\right)
_{\parallel }$ $=(x-x^{\prime },\;y-y^{\prime })$ and define $$G _{\;\nu }^{\mu }\left( \mathbf{x} , \mathbf{x} ^{\prime }\right) = 4\pi
\int \frac{d^{2}\mathbf{p}}{\left( 2\pi \right) ^{2}}e^{i\mathbf{p}\cdot
\left( \mathbf{x}-\mathbf{x}^{\prime }\right) _{\parallel }} g_{\;\nu }^{\mu
}\left( z,z^{\prime }\right) ,  \label{RedGreenDef}$$where $\mathbf{p}=(p_{x},p_{y})$ is the momentum parallel to the plane $%
\Sigma $ [@CED]. In Eq. (\[RedGreenDef\]) we have suppressed the dependence of the reduced GF $g_{\;\nu }^{\mu }$ on $\mathbf{p}$.

Due to the antisymmetry of the Levi-Cività symbol, the partial derivative appearing 