what can be done if son refuses to accept his mentally ill mother

If the relationship becomes too emotionally draining, affects your own mental health negatively, or if there's a pattern of toxic behavior that doesn't improve despite attempts to help, it may be time to consider walking away.

It also depends on how the customs, morals, and finances of the family have been. If all 3 are in line then it is hardly a matter of question about who looks after the old parents. Even daughters can come forward to take the charge.