C560 FLT CREW EXPERIENCES AN AUTOMATIC INFLT SHUTDOWN AFTER ENG FAILS DURING APCH TO LNDG AT ZZZ.

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WHILE BEING VECTORED FOR A VISUAL APCH TO RWY 10L AT ZZZ; WE WERE INSTRUCTED TO DSND TO 4000 FT ON A R BASE LEG AND TO SLOW OUR AIRSPD TO 180 KTS. WE WERE APPROX 8 MI OUT AND INFORMED BY ATC THAT THERE WAS TFC AHEAD OF US AT OUR 1 O'CLOCK POS; ALSO PROCEEDING IN FOR LNDG ON RWY 10L. WE WERE INSTRUCTED BY THE ZZZ TWR CTLR TO SQUARE OFF OUR BASE LEG TO PROVIDE SEPARATION; AND BEING THE PF; I REDUCED THE PWR TO IDLE ON BOTH ENGS WHILE SQUARING OFF OUR BASE LEG. AFTER REDUCING PWR; I OBSERVED THE OIL PRESSURE LOW ANNUNCIATOR ILLUMINATE PERTAINING TO THE L (#1) ENG; AND THE FLASHING ILLUMINATION OF THE MASTER WARNING ANNUNCIATOR LIGHTS; AS WE BOTH OBSERVED THE L (#1) ENG FLAMEOUT AND ALL ASSOCIATED ENG INST INDICATIONS AND ANNUNCIATORS CONFIRMING THE ENG FAILURE. PRIOR TO DEPRESSING THE MASTER WARNING INDICATOR; THE PNF AND I CONFIRMED THE ENG FAILURE INDICATIONS AND I PROCEEDED TO PERFORM THE MEMORY ITEMS PERTAINING TO AN ENG FAILURE WHILE ON AN APCH. AFTER LEVELING OFF AT 4000 FT AND CONFIRMING THE LOCATION OF THE ACFT AHEAD OF US; I TURNED FINAL AND PROCEEDED TOWARD THE ARPT. ATC CLRED US TO LAND AND I CALLED FOR THE CHKLIST TO CONFIRM THAT ALL MEMORY ITEMS WERE COMPLETED. AS WE PROCEEDED TOWARD RWY 10L; I CALLED FOR THE LNDG GEAR TO BE EXTENDED AND FOR ALL LNDG CHKS. I THEN ASKED THE PNF TO INFORM THE TWR THAT WE WERE DECLARING AN EMER DUE TO THE ENG FAILURE. THE TWR CTLR ASKED US IF WE REQUIRED THE EMER EQUIP TO STAND BY AND THE PNF RESPONDED AFFIRMATIVE. WHILE FLYING THE APCH TO THE RWY; AT NO TIME DID WE CONDUCT ANY OTHER CHKLIST ITEMS PERTAINING TO THE ENG FAILURE AND I DID NOT BRING THE THROTTLE INTO THE CUT-OFF POS ON THE FAILED ENG. I INSTEAD FOCUSED MY ATTN ON FLYING THE SINGLE ENG APCH AND THE PNF PROCEEDED WITH ALL ASSOCIATED CALLOUTS AS PER SOPS. WHEN I DETERMINED THAT LNDG WAS ASSURED I CALLED FOR FULL FLAPS AND WE PROCEEDED TO LAND WITHOUT INCIDENT. SHORTLY AFTER TOUCHDOWN WHILE DECELERATING AND APPLYING BRAKES; SPD BRAKES AND DEPLOYING THRUST REVERSERS ON THE OPERATING ENG; THE PNF OBSERVED THE ILLUMINATION OF THE L ENG BOOST PUMP ON ANNUNCIATOR; AND CONFIRMED THAT THE L ENG HAD BECOME OPERATIONAL. AFTER EXITING THE RWY WE WERE ESCORTED BY AN EMER VEHICLE. I INFORMED THE PNF TO INFORM THE TWR CTLR THAT NO FURTHER ASSISTANCE WAS REQUIRED AND HE DID SO. I NOTICED THE ODOR OF FUEL IN THE CABIN AFTER EXITING THE RWY AND PROCEEDED TO SELECT THE L BOOST PUMP TO THE OFF POS. WE PROCEEDED TO PERFORM ALL AFTER LNDG CHKLISTS AND CONTACTED GND FOR CLRNC TO TAXI TO THE FBO RAMP. WE TAXIED TO THE RAMP AND AFTER PARKING; SHUT THE ENGS DOWN AND PROCEEDED TO OBSERVE NORMAL ENG INDICATIONS AND PERFORMED THE APPROPRIATE SHUTDOWN CHKLISTS. THE PAX DISEMBARKED THE ACFT AND PROCEEDED INTO THE FBO BUILDING WITHOUT ANY CONCERN REGARDING THE EVENT. MAINT CHKED RIGGING FOR THE ACFT PWR LEVERS AND FOUND NO FAULT WITHIN THE SYS. THEY ALSO PERFORMED AN OPERATIONAL CHK ON THE L ENG AND THE ENG OPERATED NORMALLY. THE FUEL CTL UNIT WAS REMOVED AND REPLACED.