Verkerhsverschäft: Debt question unresolved and UG has no insurance cover

Hello dear community, I had at a gas station two weeks ago a small accident with my girlfriend’s car (Bagatelle; both vehicles have a 5cm scratch each on the bumpers). Police we did not call and exchanged insurance cards. We agreed that everyone would report his damage to the other insurance. Now I got a letter from his insurance company that the vehicle of the accident opponent at the time of the accident did not have any insurance However, a letter from his lawyer regarding the settlement of damages is available from my insurance company. The problem is that the scoundrel question has not yet been resolved, since each of us believes that the other is to blame. What happens now, because the UG had no insurance cover?