Gauge invariant Green's function for a point particle

Question

This question is a follow up to the question (Gauge invariant Green's function for electrodynamics). It is not possible to generally solve the eqution \begin{equation} \square A^{\mu}-\partial^{\mu}\left(\partial_{\nu} A^{\nu}\right)=\frac{4 \pi}{c} j^{\mu} \end{equation} However, if we specify the current to the current of a point particle, is there a general solution to \begin{equation} \square A^{\mu}-\partial^{\mu} (\partial_{\nu} A^{\nu})=\frac{4 \pi}{c} \int_{-\infty}^{\infty} d s v^{\mu}(s) \delta^{4}[x-z(s)]~? \end{equation}

Answer 1

This argument is still valid, for this expression of $j^\mu$ as for any other current distribution.

The reason is that the LHS is invariant under $A^\mu \rightarrow A^\mu +\partial^\mu f$ for any function $f$, so there is no hope of find a general solution without fixing the gauge.