What was found
FROM KSAT TO MYNN WHEN WE EXPERIENCED AN ENGINE FAILURE IN FLIGHT. AT FLIGHT LEVEL 390 APPROXIMATELY 100 MILES WEST OF THE SOUTHERN COAST OF FLORIDA, THE LEFT (#1) ENGINE ROLLED BACK QUITE ABRUPTLY IN FLIGHT. WE DECLARED AN EMERGENCY DUE TO THE LOSS OF THRUST AND THE CABIN PRESSURE RISING SINCE ONLY 1 ENGINE WAS SUPPLYING PRESSURIZATION. UPON OUR DECENT DOWN TO 10,000 FEET, WE COMMENCED OUR SOPS AND RUN THROUGH OUR EMERGENCY CHECKLIST. THE FIRST ONE CALLS TO SECURE THE ENGINE AND TAKE IT OFFLINE. AFTER COMPLETING THIS WE ARE TO MAKE THE DECISION TO TRY AND RESTART IT. BECAUSE THERE WAS NO INDICATION OF FIRE, FOREIGN OBJECT DEBRIS, OR FROZEN FAN (N1 OR N2), WE DECIDED TO RESTART THE ENGINE. AFTER RUNNING THROUGH THE PROPER CHECKLIST TO RESTART THE ENGINE, WE COULD NOT GET THE ENGINE TO LIGHT OFF. SO, WE DECIDED TO RESECURE THE ENGINE AND CONTINUE TO ON TO OUR DIVERSION AIRPORT, KRSW, WITH ONE ENGINE AND LAND AS SOON AS POSSIBLE. WE CONTINUED WITH THE "SINGLE ENGINE APPROACH TO LANDING" CHECKLIST AND LANDED THE PLANE AT KRSW. THIS IS A 12,000FT RUNWAY AND AFTER RUNNING THE LANDING DISTANCE WITH PARTIAL FLAPS WE CONCLUDED IT WAS MORE THAN LONG ENOUGH. THE LANDING ITSELF TURNED OUT TO BE A NON-EVENT AND WE WERE ABLE TO SAFELY TAXI TO THE FBO AND SAFELY DE-PLANE THE PASSENGERS.