Icelandair used to fly between Luxembourg and Keflavik and onwards to New York Idlewild (now JFK) for appr. 44 years. (BTW, Icelandair was the product of the merger between two icelandic airlines in the early 1970's, the carrier which started service between LUX and IDL/JFK was actually Loftleidir).
They started in 1955 with the DC-4, later changed to the DC-6. From June 1964 they operated the Canadair CL-44D4 and after a while the stretched CL-44J which seated over 180 passengers (!). Loftleidir called the CL-44 the "Rolls Royce 400" due to their engines. At that time, Loftleidir flew LUX-KEF-JFK 5 times a week with a mixture of DC-6 and CL-44, plus operating several, less frequent, DC-6 flights from London, Amsterdam and Scandinavia via Keflavik to JFK. BTW, the CL-44 saved 4h 30min between JFK and KEF compared to the DC-6. On the sector KEF-LUX the time saving was another 1h50min! In the late 1960's LUX-KEF-JFK was served daily by CL-44's while London and all the other European destinations were only served once or twice a week. At that time Loftleidir started to set up a subsidiary called "International Air Bahama" which flew charter flights from Luxembourg via Shannon to Nassau, using DC-8 equipment.
These years also saw the replacement of the CL-44 by the DC-8 on Loftleidir's main line. The CL-44 were converted to freighters and handed over to Cargolux, where Loftleidir held a stake of 33%.
In the late 1970's the airline which was now called "Icelandair" put the DC-10 on a 6x weekly service LUX-KEF-JFK while the DC-8 served Washington and Chicago ex Keflavik, with good transfers to the DC-10 flight. Unfortunately the forced grounding of all DC-10's after the American Airlines Crash in Chicago 1979 put a quick end to the DC-10 operation at Icelandair. So in the 1980's the DC-8-63 was found on a daily LUX-JFK flight, to be replaced in the early 1990's by the 757 which remained on this route until Icelandair ceased operations into LUX in 1999.
The original reasons for their presence in LUX were as far as i know indeed the traffic rights. Only from LUX they had the chance to undercut the high IATA rates in the 1950's and 60's to offer some kind of "budget air bridge" between Europe and the USA. They had organized bus shuttles fom other European cities to LUX to increase the catchment area of this small country.
Even in the 1980's I remember that an Icelandair from LUX to JFK including the bus shuttle from Germany to LUX was remarkably cheaper than a flight FRA-JFK on LH or PA. As far as I know they had no commercial agreement with Luxair though....
When the other airlines started to decrease their fares on the Atlantic, the passengers started to look for more comfortable nonstop flights rather than for the transfer flights and bus shuttles via LUX. Therefore the operation in LUX could no longer be kept at a satisfactory level. Therefore Icelandair decided in 1999 to stregtehn their presence in Paris and Frankfurt by withdrawing the daily rotation from LUX in favor of increased service to these two destinations.
If anyone reached this point of my long posting I hope that I could at least answer some questions.