Is the amount paid by the son to his mother, after his brother refused to participate in it, considered a debt upon the brother, or is it waived from him because he believes that the mother possesses wealth and does not need assistance?

Allah and His Messenger have made it obligatory to honor one's parents and treat them with kindness, especially in their old age, and special care is given to the mother. It is also obligatory for a child to financially support their poor parents if they are wealthy. Withholding financial support from the mother due to a dispute between them is not an act of righteousness and is considered disobedience. The son must strive to appease his mother, apologize to her, and resume her financial support. If the brother continues to withhold financial support, then you should seek your reward from Allah for what you spend on your mother and strive to reconcile between them.