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The official airline of Ireland is Aer Lingus.
and what about Ryanair - that's the original Irish low cost airline.
Hawaiian Airline
It means the airline pays money to the Chicago Cubs to be its "official airline" and basically get advertisement from the Cubs
The United States does not have an official airline.
Continental Airlines is not the official airline of any country, though it is one of the largest carrier in the US and the world.
JAL