There are a million web sites for this. I often look at flightstats.com to see the arrivals or departures for today, yesterday, and tomorrow. This finds all the low-cost airlines that don't publish their fares. But it doesn't work well when looking at a route that's not daily or for planning a trip in a different season.
Otherwise you would have to look at the airline's timetables which I don't think are aggregated by airport anywhere.
But on matrix.itasoftware.com you can do the same search you're doing on kayak but force it to return only direct flights. I searched for SXM to LHR,CDG,ZRH with a routing code of "F" which means one direct flight. The only result was AF3515. You would have to double check that this is a non-stop but it's definitely direct.
Alternatively if you search for SXM to BOM,DEL with a routing code of "F F" (precisely two flights, though they might actually have stops but one flight number) then it gives two options. SXM-CDG-BOM/DEL on AF or SXM-EWR-BOM/DEL on UA. The UA option is a grueling 15 hour flight though.