Eq. (\[RedGreenFuncA1\]) demands the derivative of $\mathfrak{g}$ to be discontinuous at $z=z^{\prime}$, *i.e.,* $\partial _{z} \mathfrak{g} \left( z ,
z ^{\prime} \right) \big|  _{z= z^{\prime -}} ^{z=z^{\prime +}} = -1$, and then the continuity of $\mathfrak{g}$ at $z=z^{\prime}$ follows [@CED].

Now we observe that Eq. (\[RedGreenFunc\]) can be directly integrated by using the free GF Eq. (\[RedGreenFuncA1\]) together with the properties of the Dirac delta-function, thus reducing the problem to a set of coupled algebraic equations, $$g_{\;\ \sigma }^{\mu }\left( z,z^{\prime }\right) =\eta _{\;\sigma }^{\mu }%
\mathfrak{g}\left( z,z^{\prime }\right) -i\tilde{\theta}\epsilon _{\;\;\;\;\
\nu }^{3 \mu \alpha }p_{\alpha }\mathfrak{g}\left( z,a\right) g_{\;\ \sigma
}^{\nu }\left( a,z^{\prime }\right) .  \label{RedGreenFuncA3}$$ Note that the continuity of $\mathfrak{g}$ at $z=z'$ implies the continuity of $g_{\;\
\sigma }^{\mu }$, but the discontinuity of $\partial _{z} \mathfrak{g}$ at the same point yields $$\partial _{z} g_{\;\ \sigma }^{\mu } \left( z , z^{\prime }\right) \big| %
_{z= a ^{-}} ^{z = a^{+}} = -i \tilde{\theta}\epsilon _{\;\;\;\;\ \nu }^{3 \mu
\alpha }p_{\alpha } \partial _{z} \mathfrak{g}\left( z,a\right) \big| _{z=
a^{-}} ^{z = a^{+}} g_{\;\ \sigma }^{\nu }\left( a,z^{\prime }\right) = i \tilde{%
\theta}\epsilon _{\;\;\;\;\ \nu }^{3 \mu \alpha } p_{\alpha } g_{\;\ \sigma
}^{\nu }\left( a,z^{\prime }\right) ,  \label{BCRedGreenFunc}$$ from 