Learjet 45 Captain reported a left engine warning light followed by loud boom; aircraft yaw and vibration while in cruise. The flight crew requested priority handling and continued to destination airport.

Narrative

1. L ENGINE CHIP white Crew Alert System at FL340 in VMC After confirming no action was required by QRH (white advisory message); continued climb to FL430. 2. Loud boom and aircraft yawing and light shaking at FL430 IN VMC (approximately 145 miles outside of departure point ZZZ and 85 miles to ZZZ1) 3. Reduced power on both engines and depressed control wheel master to maintain direction control. ATC was informed of the situation. Both crew members cross checked and confirmed it was the left engine was the issue. 4. Upon reducing the left engine to idle; Aural Warning 'Left Engine Fire' and accompanied by light smoke in cockpit. Oxygen mask were donned by both crew members with 100% oxygen selected and crew communication was established. 5. Memory items completed and left engine was secured. 6. No more left engine fire indication and smoke dissipated After confirming visually that smoke was not visible crew members removed oxygen mask one at a time. 7. Continued with emergency checklist and descent in VMC8. Entered IMC at around 5;000 feet; temperature well above freezing (above 10 degrees centigrade); and proceeded with single engine ILS into ZZZ1. 9. Single engine landing at ZZZ1 on Runway XXR and taxi to signature under own power.