in the second term of the GF Eq. ([EqsGreenMatrix]{}) does not introduce derivatives with respect to $z$, but only in the transverse coordinates. This allows us to write the full reduced GF equation as $$\left[ \partial ^{2}\eta _{\;\nu }^{\mu }+i\tilde{\theta}\delta \left(
z-a\right) \epsilon _{\;\;\;\;\ \nu }^{3 \mu \alpha }p_{\alpha }\right]
g_{\;\ \sigma }^{\nu }\left( z,z^{\prime }\right) =\eta _{\;\ \sigma }^{\mu
}\delta \left( z-z^{\prime }\right) ,  \label{RedGreenFunc}$$where $\partial ^{2}= \mathbf{p} ^{2}-\partial _{z}^{2}$, $p^{\alpha
}=\left( 0 , \mathbf{p} \right) $ and $\mathbf{p} ^{2} = - p ^{\alpha} p
_{\alpha}$. The solution to Eq. (\[RedGreenFunc\]) is simple, but not straightforward. For solving it we employ a method similar to that used for obtaining the GF for the one-dimensional $\delta$-function potential in quantum mechanics, where the free GF is used for integrating the GF equation with $\delta$-interaction. To this end we consider a reduced free GF having the form $\mathcal{G}_{\;\nu }^{\mu }\left(
z,z^{\prime }\right) =\mathfrak{g}\left( z,z^{\prime }\right) \eta _{\;\nu
}^{\mu }$, associated with the operator $\partial ^{2}$ previously defined, that solves $$\partial ^{2}\mathcal{G}_{\;\nu }^{\mu }\left( z,z^{\prime }\right) =\eta
_{\;\nu }^{\mu }\delta \left( z-z^{\prime }\right), \label{RedGreenFuncA1}$$satisfying the standard boundary conditions at infinity, where $$\mathfrak{g}(z,z^{\prime })=\frac{1}{2p}e^{-p|z-z^{\prime }|} \label{RFSGF}$$ and $p = \vert \textbf{p} \vert$. Note that 