Most regional aircraft makers provide only one engine option for their aircrafts, like Rolls Royce for BAe Jetstream 146, Embraer Jets, Fokker Jets, GE for Canadair CRJ. The same also goes for most mainline single aisles, like Boeing B737 (only PW JT8D on B737-100/-200, DC-9, MD80 series, V2500 on MD90, CFM56 on B737 2nd generation & B737NG.)
But there are more than one engine option for most widebody jets & Airbus A320, except for jets with special requirement like Boeing B777-200LR & -300ER requiring 110,000lb engine thrust & solely powered by GE90.
Because of limited engine options offered on most aircrafts and the co-operation of some airlines with certain engine manufacturers, many airlines specify aircrafts with a brand of engine, or even (not) introducing a certain type of aircraft.
Swire Group owns Cathay Pacific, Dragonair & HAESL of 45% joint venture with Rolls Royce. So Cathay Pacific & Dragonair specify all aircrafts with Rolls Royce engines (except Dragonair A320 of V2500 engines with Rolls Royce technology & Cathay Pacific A340-300, which has only CFM56 engine option.) This also makes Cathay Pacific chooses Rolls Royce-powered L-1011 of shorter range & less economy instead of DC-10 & MD11, which have no Rolls Royce engine option, and refused to introduce B767 for routes with fewer passengers due to lack of Rolls Royce engine option except for -300 variant.
The same also goes for Air Canada, Virgin Atlantic & BMI, which uses only Rolls Royce-engined Fokker & Embraer regional jets, and would be changed to full Rolls Royce-powered Airbus & Embraer fleet.
Some countries with more than one major carriers, like Japan, South Korea & Taiwan have all major carriers use the same brand of engines, like Pratt Whitney for JAL, ANA & JAS of Japan, GE for Korean Air, Asiana of Korea, China Airlines & EVA Air of Taiwan. In fact EVA Air & Korean Air chooses Boeing B777-300ER instead of Airbus A340-500/-600 just because of GE engine option, even B777 is much less reliable than A340 because of twin engine design.
The European airlines have very diversed engine selections. British Airways uses mainly Rolls Royce engines, but it also uses GE engines on some B747 & B777. Air France uses nothing but GE engines & CFM engines, a 50/50 joint venture between GE & Snecma of France. Lufthansa uses CFM engines on A320, but V2500 engines on A321, and is very rare in the world.
Swiss carriers do not adhere to an engine brand, and they choose engines just based on environmental performance, because of Swiss high degree concern on air quality & hence penalties on high emission engines. So they use only CFM56 single aisle engines & Rolls Royce Trent engines on A330 & A340-600.
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