Gulfstream IV / G350 / G450
Air Traffic Control Fully Certified
Situational Awareness
Aircraft Equipment Problem Critical; Conflict Airborne Conflict
General Declared Emergency
Aircraft
G-IV was descending from FL240 to cross ZZZZZ Intersection at 13;000 feet en route to ZZZ. A320 was at FL270; landing ZZZ1; requesting lower [altitude] because of moderate chop. I observed G-IV descending through FL231 to make the ZZZZZ restriction; then I issued A320 FL240. A320 started a descent. About thirty seconds later; G-IV climbed back to FL234. I advised G-IV that he must continue descent and not climb. G-IV acknowledged my transmission; and about thirty seconds later declared an emergency - right engine flame-out; five souls on board; requested direct to airport. In the same situation I would tell A320 that they would have to vector off course to get lower sooner. If the pilot agreed; I would vector east of the ZZZ1 arrival to avoid the emergency aircraft and get A320 lower as soon as possible.
En route Controller described a loss of separation event when an inbound aircraft declared an emergency and failed to adhere to descent instructions.