hi, I´d like know if a no american kid with her no american mother travelling together without father, needs any authorization?
I do not know the rules for every country, but I believe in most places if one parent is traveling out of the country with their child they need either authorization from the other parent or a legal document showing that they are the sole guardian of the child, if that is the case.
It is always better to have these documents and not need them, then to not have them and need them and risk something messing up your vacation.
It is probably best to check with Royal Caribbean directly, or verify the rules of your particular country. Sorry I can not be more help.