Thank you for the information and your question and I am sorry to read about your mother's accident. I hope she is improving. Your mother has the right to hire a traffic law attorney to fight the ticket. If the facts support the statement you have made about them, it should be fairly easy for her attorney to get the ticket dismissed. That often changes the determination of liability for the insurance company. However, they are no bound, one way or the other, by whether the police issued a citation. They conduct their own investigation and interview all parties, while also looking at the damage to the vehicle to come to a liability decision.
So she, or her attorney, can contact the insurance company to insure they know all of the facts and have access to the vehicle, etc., and let them know they will be disputing the ticket. If the insurance company ultimately makes a liability decision against her anyway, she doesn't have many options other than to complain to the Department of Insurance or file a bad faith claim against the insurance company, which would be difficult to win unless there is proof they ignored obvious evidence.
As the for police officer, it is unlikely that there would be any action, other than a complaint to the police department, that there would be a suit here against the police department (can't sue the police officer under the law). In other words, this report was likely just his perception of what occurred and what he heard that day, which is technically not a "false report." Just one with errors, which again, her attorney can fight in court. Unless there is evidence that this police officer is discriminatory against certain ethnic groups or took some sort of a payoff, this is not an issue that would support any sort of suit.
Please let me know if you need any clarification. If not, then if you can take a moment to leave a positive rating in the box above, I will receive credit for assisting you today. Thank you