I believe mantener in this case implies financial/living support, whereas apoyar would mean that she's cheering him on, or got his back, or somesuch.
What about "She gives support to her brother." Is that wrong? Also, does support mean all of the various 'supports' that you could mean in English? Like, money, emotional support, actually supporting someone who is falling over, etc?
why and under what circumstances does the ' a ' appear after the word mantiene.
I know this one :-) This is the famous "personal a"... it seems to be a pattern in Spanish that when a verb is acting upon a person, the word "a" is inserted after the verb... if you're trying to translate to English there's no need to translate the "a" so essentially it's silent, but it just needs to be there. It's very specific to people though... the same verb applied to an inanimate object or even an animal doesn't need the "a". I can't think of any examples off the top of my head, but trust me, you'll see this a lot!