Recently I was calculating some stuff in curved (ADs to be exact) spaces, when the following question came to my mind, Suppose you have in general a differential operator $\hat{D}$ acting on the metric $g_{\mu\nu}$ such that, $$\hat{D}g_{\mu\nu} = T_{\mu\nu}$$ where $T_{\mu\nu}$ is a stress tensor (just an indexed space-time function for now).
Question: What are the necessary conditions for the Green's function of $\hat{D}$ to be translation invariant, that is $G(x,x') = G(|x-x'|)$?
As far as my knowledge goes, translation invariance of the Green's function corresponds to an invariance/symmetry of the operator $\hat{D}$ itself. What practical checks do I perform on $\hat{D}$ to see if it is translation invariant ? I tried to solve it but made absolutely no progress, nor found any relevant literature. Can someone help by pointing out relevant literature and/or explaining a procedure to check the translation invariance?