Date: 201901
Local time of day: 1201-1800
Location: ZZZ.Airport, US
Flight conditions: VMC
Weather elements / visibility: 10
Light: Daylight
Aircraft: Learjet 60
Operator: Air Taxi
Operating under: Part 135
Flight plan: IFR
Mission: Passenger
Flight phase: Cruise
Airspace: Class A ZZZ
ATC / advisory: Center ZZZ
Aircraft component: Turbine Engine
Component problem: Malfunctioning
Reporter organization: Air Taxi
Reporter function: Captain; Pilot Flying
Reporter qualification: Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Air Transport Pilot (ATP)
Second reporter function: First Officer; Pilot Not Flying
Anomaly: Aircraft Equipment Problem Critical
Detector: Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Issued New Clearance; Flight Crew Landed As Precaution; Flight Crew Inflight Shutdown; Flight Crew Diverted; Flight Crew Requested ATC Assistance / Clarification; General Maintenance Action
Contributing factors: Aircraft
Primary problem: Aircraft

Narrative: While in normal cruise at FL400 a low fuel pressure light in the annunciator panel lit up. I immediately called for the copilot to refer to the QRH. While going through the checklist the left engine rolled back and stopped producing thrust. About 2 min between initial indication and roll back.I noticed while going through the checklist that the left jet fuel pump button on the fuel panel was pushed out but not lit up with the amber 'off' light. I tried pushing it back in but it would not stay in the indent position.After the engine rolled back the plane was unable to hold altitude; so I initiated a 1000 fpm descent while notifying ATC of the issue. While descending; I called for the Engine restart checklist; as the left engine N1 was still spinning and there was no indication of fire. While [the FO (First Officer)] looked for the checklist I turned on the left igniter and the left boost pump. The engine remained at minimal thrust.While this was happening we were coordinating with ATC a suitable airport to divert to. We decided that ZZZ was the best option due to its proximity; runway length; and indicated weather (VFR).During our descent through the mid FL200's we established visual contact with the airport and I decided to conduct a single engine landing visual to runway XX. I called for the engine shutdown checklist. During the checklist; we noticed a loud hum or Whine from the rear of the airplane; presumably the left engine. We finished the checklist and secured the engine. The noise went away.ATC switched us over to Unicom after we confirmed that we had the runway made. I then called for the single engine landing checklist. I over flew the airport and entered a left base.After landing we established contact to ATC through an [air carrier] flight and confirmed we were on the ground with no complications. We then confirmed with personnel on the ground that there was no external indications of fire. We then taxied to the FBO and shut down the aircraft.


Second narrative: [Report narrative contained no additional information.]


Synopsis: LJ60 flight crew reported diverting after an inflight engine shutdown following a low fuel pressure warning.