What is the ruling regarding a son traveling abroad for work to secure a livelihood for his family, given his elderly and only mother's refusal of his travel due to her intense love for him and her need for his service and care, even though there are other siblings who attend to her needs?

If you are in need of travel and would be harmed by not traveling, then there is no sin upon you in traveling without your mother's consent. Safe travel for trade and other purposes is permissible so that one's livelihood is not cut off. However, it is advised that you strive to convince your mother of your need to travel and satisfy her by all lawful means. If she does not consent, then be patient and seek reward from Allah, and strive to earn a lawful livelihood in your own country. You must be dutiful and good to her, for honoring one's parents is among the best acts of closeness to Allah and a cause for His pleasure and success.