Chris1979 - angatol has kindly listed the fare code and information. Those particular prices I've listed were based on the second weekend in August travelling and returning midweek (ie. cheapest days and with a Saturday night stay). There may be cheaper prices available at other times given it appears to be running in Nov and Dec/Jan.
I've managed to get those prices (give or take £20) including LCY-JFK, showing up both on BA and ITA matrix, albeit I guess you may want to book via AA for the full TPs depending on your plans.