^{\prime} \right)$ in Eq. (\[gjS\]). The result is $$g ^{j} _{\; \sigma} \left( z , z ^{\prime} \right) = \eta ^{j} _{\; \sigma} 
\mathfrak{g} \left( z , z ^{\prime} \right) + i \epsilon ^{3 j k} _{\;\;\;\
0} p _{k} \left[\eta ^{0} _{\; \sigma} - i \tilde{\theta} \epsilon ^{3 0 i}
_{\;\;\;\ \sigma} p _{i} \mathfrak{g} \left( a , a \right) \right] A \left(
z , z ^{\prime} \right) .  \label{gjS-2}$$ Equations (\[g0S-3\]) and (\[gjS-2\]) allow to write the general solution as $$g_{\;\nu }^{\mu }\left( z,z^{\prime }\right) =\eta _{\;\ \nu }^{\mu }%
\mathfrak{g}\left( z,z^{\prime }\right) +A \left( z,z^{\prime }\right)
\left\lbrace \tilde{\theta}\mathfrak{g}\left( a,a\right) \left[ p^{\mu}
p_{\nu } + \left( \eta _{\;\ \nu }^{\mu } + n^{\mu }n_{\nu } \right) p ^{2} \right] +i \epsilon _{\;\ \nu }^{\mu \;\ \alpha 3}p_{\alpha}
\right\rbrace ,  \label{GenSolGreenPlaneConf}$$where $n _{\mu} =\left( 0,0,0,1\right) $ is the normal to $\Sigma $ The reciprocity between the position of the unit charge and the position at which the GF is evaluated $G_{\mu \nu }(\mathbf{x},\mathbf{x}
^{\prime }) = G_{\nu \mu }(\mathbf{x} ^{\prime},\mathbf{x})$ is one of its most remarkable properties. From Eq. ([RedGreenDef]{}) this condition demands $$g_{\mu \nu }(z,z^{\prime }, \mathbf{p})=g_{\nu \mu }(z^{\prime },z,- \mathbf{%
p}),$$which we verify directly from Eq. (\[GenSolGreenPlaneConf\]). 