Does there exist a green's function that does not have translation symmetry? I noticed that most Green's functions I have used take on the following functional form $G(x_1,x_2)=G(|x_1-x_2|)$. I assume these subsets of Green's functions are translationally invariant? Correct me if I am wrong.
Does there exist a green's function that does not have translation symmetry?
For example $G(x_1,x_2)=G(x_1x_2)$?

Yes, take any fundamental solution of a PDE which doesn't have translational symmetry, ex: the fundamental solution of $Ly=xy', y'(1)=0\,.$ If $Ly=f\,, f(1)=0$ then $\displaystyle y(x)=\int_{-\infty}^\infty \chi_{[1,x]}(y)\frac{f(y)}{y}\,dy\,.$ So the fundamental solution is $E(x,y)=\frac{\chi_{[1,x]}(y)}{y}$ which isn't translationally invariant.
