Why lie? That only causes problems, possibly up to insurance fraud. I wouldn't want to try explaining how the car was stolen but now is back without any help.

Sinshine, it wouldn't actually be lying. If the mother doesn't own some portion of the car, each time she drove it, she was in fact committing car theft. It doesn't stop being a crime just because the thief brings it back. If OP lives somewhere that it's a crime to drive without a valid insurance plan, and the mother wasn't on OP's plan as a driver of that vehicle, that's another crime.

Unless OP's mother has partial ownership of the car, and also is on OP's insurance plan or has the car on her insurance plan, if they can prove that they weren't in the country, they don't have to pay the fee. If they weren't in the country to drive it, they obviously didn't commit the driving violations, and thus shouldn't be able to be charged. If the mother doesn't have any ownership of the car, each time she drove it, she was in fact committing car theft. Given that I believe this same poster mentioned the mother stealing several thousand dollars from one of her own children, in OP's shoes, I wouldn't just prove I was out of the country, but also inform the judge that the incidents cited occurred during times that the car had been stolen. For a person as shitty as the mother is, there really isn't anything to lose by getting her arrested.