Well...

LAX-LHR: UA/AA both offer service
LAX-SFO: UA/AA both offer service
LAX-SMF: UA only
SFO-CIC: UA only
LAX-LAS: UA/AA both offer service
SFO-LAS: UA only (though AA does fly SJC-LAS)
LAX-JFK: UA/AA both offer service
SFO-JFK: UA/AA both offer service
LAX-MIA: only AA flies nonstop
SFO-MIA: only AA flies nonstop

Just so that you can get a sense of each carrier's schedules, on a random date (Friday, July 15, 2005) UA/AA offer the following frequencies (nonstop flights only):

LAX-LHR: UA 1X, AA 2X
LAX-SFO: UA 15X, AA 7X
LAX-SMF: UA 7X, AA N/A
SFO-CIC: UA 3X, AA N/A
LAX-LAS: UA 7X, AA 3X
SFO-LAS: UA 8X, AA N/A (AA SJC-LAS 1X)
LAX-JFK: UA 6X, AA 12X
SFO-JFK: UA 7X, AA 5X
LAX-MIA: UA N/A, AA 5X
SFO-MIA: UA N/A, AA 3X

Hope this helps!