A LEAR 24 IN CLB AT FL405 HAD THE R ENG FLAME OUT. DSNDED TO FL290 FOR RELIGHT. ENG RELIGHTED AND RETURNED TO FL390 WITHOUT FURTHER PROBS.

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LEAR 24 ENRTE FROM ZZZ TO ZZZ1 CLBING THROUGH 40500 FT FOR FL410. HEARD A 'POP' AND R ENG INSTRUMENTS INDICATED FLAME-OUT (IE NO EGT; WINDMILLING RPM; AND REDUCED OIL PRESSURE). NO OTHER ADVERSE INDICATIONS OF ENG SEIZURE OR FIRE. REQUESTED IMMEDIATE DSCNT TO FL240 (FOR ENG RESTART ENVELOPE). SOME DELAY FROM THE CTR DUE TO TFC. I ELECTED NOT TO DECLARE AN EMER YET; AS THINGS SEEMED 'OTHERWISE NORMAL;' BUT DID NOT WANT TO DELAY DSCNT AS MAINTAINING ALT WOULD EVENTUALLY NOT BE POSSIBLE. CLRED TO FL290 WITH A DELAY AT THAT ALT FOR APPROX 3 MINS. CTLR ASKED WHY WE WANTED DSCNT. I TOLD HIM IT WAS NECESSARY FOR RESTARTING ENG. RECEIVED DSCNT TO FL240. PASSING FL295 RESTARTED ENG. ALL INDICATIONS NORMAL. CLBED BACK TO FL390 (AFTER CLRNC RECEIVED). MAINT INVESTIGATING CAUSE OF FLAME OUT. (FOD; STATOR AREA MISMATCH; FUEL CTL?) CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR STATED THE AIRPLANE HAS NOT BEEN WORKED ON BUT IS AWAITING A BORESCOPE OF THE STATOR VANES IN THE COMPRESSOR. IF FOUND OK THEN THE ENG FUEL CTL WILL BE REPLACED. THE RPTR STATED THAT THE SAME ENG FLAMED OUT 40 DAYS AGO.