Car + freedom of claims discount inherited —important question about this—

Hello, my father passed away in December and left me his car (but without a will). Now I went to his insurance company and they told me no problem it will be my name in the insurance policy and the damage allowance is mine. Only my mother would have to sell the car to me (obligatory for 1€) so that I can rewrite the letter. Since my mother is not the holder, I must have any proof (death certificate, double card) for the Zula Do I need the double card of the insurance company first? What is the order, first insurance and then registration office, right? Unfortunately, I did not reach anyone at the registration office by phone today. Who can help me?