Was researching a summer trip today and discovered some really great fares from LAX to South America.

Lots of availability and some variety in destinations. Some examples that I found:

LAX-MIA-VVI (Santa Cruz, Bolivia) for $1,400 r/t
This flight stops in La Paz on the way outbound, return is nonstop VVI-MIA; operated by 757-200 (lie-flat)

LAX-ASU (Asuncion) about $1,700 r/t, LAX-MIA-ASU and return same, operated by 757-200 (lie-flat) MIA-ASU

LAX-EZE/AEP on LAN/TAM for $2,100 r/t, variety of routing options including LAX-JFK-GRU-AEP

LAX to SCL from $2125 on LAN

LAX to GRU from $2133 on LAN/TAM

I am headed to EZE in July so that was my basis for the date research but you can find a variety of really great deals. Found availability pretty much widespread and also available from other origin cities (SFO, for example). Any OTA, AA.com or LAN.com can show and ticket these flights.