"LAN" as often used refers to Lineas Aereas Nacionales de Chile, previously LAN Chile, or Lan Chile. Lan Perú is a subsidiary, and is (also) elegible for oneworld benefits. Lan Ecuadór, forget about it, regardless of the logo on the aircraft fuselage, ditto Lan Dominicana. What hand is the boarding pass in?
As for AeroContinente, they were refused insurance, therefore grounded. The Latin American media is full of interesting aspersions about its CEO and the alleged drug trafficking. True? Maybe, maybe not... but I'd certainly not be surprised. Politics? Most assuredly.
Some airlines in the southern nations have stunk to high heaven, allowed to function only because of some "fix." A classic example was the deservedly and now-defunct LANSA Perú. My cousin Flint died thanks to their poor maintenance and operations (according to the Peruvian government report, the L188-A was overloaded, and the captain committed major mistakes when the No. 3 engine quit - out of Cusco.) It wasn't the first, not even the first Electra, they lost.
Of course, in Ecuadór, we have Icaro - would you fly on an airline named after a guy who flew to high, melted the wax on his wings and crashed into the sea? (Or had its safety record?)
LAN Chile seems comfortable and has good service, in my experience. Be cautious. Choose wisely. Fly well.