unfortunately, only two airlines fly from south america to australasia nonstop- lan chile and aerolineas argentinas.

you can go the other way, though- SAA GRU-JNB-PER and NZ PER-AKL. Hell of a trip, though. GRU-ORD-LAX-AKL is only 300 miles longer. Or you could try going through europe (GRU-FRA-SIN-AKL), but that's a hell of a lot longer.

If you're buying, I'd bite the bullet and fly LAN GRU-SCL-AKL. If you're redeeming, then you have to pick your poison (I'd probably take GRU-JNB-PER-AKL over flying UA and subjecting myself to US customs and immigration).