Here is a problem the flight from JFK-FRA with SQ arrives 10:00 and the LY flight is 11:00 it may be to close, and the way back the earliest flight landing in FRA with LY is 9:55 and the flight from FRA-JFK is 8:20! so I guess only a one way is an option.
First of all, the SQ flight usually arrives an hour earlier than scheduled, so yuo might have enough time to get on the 11:00 LY flight.
Second, there's a second LY flight leaving FRA at 7:30PM.