Can you re-catch mewtwo in Pokemon Y?

Question

So I read somewhere here that if you accidentally fainted mewtwo it would re-appear if you defeat the elite 4 again. If I trade my mewtwo, will it also re-appear after defeating the elite 4?

Answer 1

No, unfortunately.

If you caught Mewtwo, it won't appear again. It doesn't matter if you trade it or not, if you were in battle with Mewtwo, and you caught it in a PokeBall, that will be the last time you battle with Mewtwo ever again. Trading it won't let another Mewtwo appear.

But yes, if you fainted it, it would appear again after beating the Elite Four a second time.

Hope I helped. :)
Sources: Experience and Logic