Proper-time flow equation and non-local truncations in quantum gravity

We study the flow of the non-local truncation in quantum gravity and we focus in particular on the Polyakov effective action for a non-minimally coupled scalar field on a two dimensional curved space. We show that it is possible to explicitly integrate the flow of all the local and non-local operator terms up to $k=0$ and recover effective action without the integration of the conformal anomaly.