Proposition: A reduced wave inverse need not solve invariant equations

There are smooth compact symmetric sources $Q_{\mu \nu }$ for which the past-zero reduced equation

has a smooth retarded solution, while

has no smooth solution. This example is not an Einstein–scalar source model.

Proof. Choose compact smooth $s$ with ${\partial }_{t}s\not\equiv 0$, and let $Q_{00}=s$, all other components zero. Then ${\partial }^{\mu }{Q}_{\mu 0}=-{\partial }_{t}s\not\equiv 0$.

The reduced equation has solution

by the retarded wave construction and ${\mathsf{r}}^2=\mathrm{id}$. If the invariant equation had a solution, the linearized Bianchi identity would imply ${\partial }^{\mu }{Q}_{\mu \nu }=0$, a contradiction.