What is the proper course of action upon discovering one's mother's affair with another man, given the father's refusal to intervene, the fear of causing divorce and its impact on younger siblings, and the apprehension of being labeled a cuckold?

You must advise your mother gently and politely, and warn her of the consequences of her actions with compassion and a desire to save her, while rejecting the evil in your heart and with your tongue so that you do not become a cuckold (dayouth). This problem might stem from a prolonged separation between the spouses, and such a separation is not permissible Islamically, nor does it align with the goals of marriage in building a family characterized by affection and mercy and providing a suitable environment for raising children. Even if the separation is with the consent of both parties, it contradicts the wisdom of marriage, and the husband's right of guardianship (qawamah) does not fall away. He must oversee the upbringing of his family and prevent them from committing sins, in accordance with the Almighty's saying: "O you who have believed, protect yourselves and your families from a Fire." If the cause of the separation is from one of the spouses, then that spouse is sinful, and the other has the right to demand their marital rights.