Begin by using paragraph spacings, logical and chronological flow in your presentation of facts, and a separation between fact, fiction, commentary, and opinion. And, please stop omitting relevant facts, of which I can already sense there are a few.

Lastly, if you were really "deported", I don't think you should be planning a vacation to Costa Rica anytime soon. If you were "refused entry" without prejudice, then say so. Even your title is tendentious.

You may begin your redemption by telling us what nationality and residency you and/or your children hold. If US citizens, you'd need a passport, period. If AA did not tell you this, or misinformed you, then shame on them, but the ultimate responsibility lies with you. I am also very surprised AA let you check-in in LGA without a passport for your child.