Urzai AU - While on a ship with the kids, a huge storm hits and the fire family’s ship is destroyed, leaving the family separated. Ozai and Ursa, having survived the storm together, wash up on the shores of the South Pole with no memory of who they are or how they got there.
Avatar au: for whatever reason (Ursa takes off with Zuko to find somewhere safe, Iroh takes him around the world, he gets banished to the pole, etc) Zuko comes to hang out at the South Pole and becomes unlikely friends with the water kids. It obviously starts out rocky, but gets there eventually.
Avatar AU - A world where the South Pole didn’t fold and Princess Katara or Prince Sokka are being married off to Prince Zuko or Princess Azula for an allience, but before their families can arrange any marriage the four are kidnapped so that the alliance doesn’t happen and they end up all the way in Ba Sing Se and have to travel home together.