PAXby
I think you will get some flights on a route with First class and other at a different time with just Business. For example EK has currently 3 flights from Manchester to DXB, but only the lunchtime arrival in Manchester and later departure has First Class. I suspect this may happen on routes from London to say New York for example with BA or indeed other European carriers that operate more than one flight a day to a destination or indeed where there is little demand.
Cheers
Mr Mac