A B767 FLC EXPERIENCES A FLAMEOUT IN THEIR #1 ENG AT FL410 WHILE DETOURING TSTM ACTIVITY W OF PXV; IN.

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WE WERE AT FL410 W OF PXV DEVIATING N OF COURSE FOR WX AND TRYING TO MAINTAIN VMC TO AUGMENT OUR RADAR AVOIDANCE. THE FASTEN SEATBELT SIGN WAS ON AND WHEN WE ENCOUNTERED LIGHT CHOP WE ALSO SAT THE FLT ATTENDANTS AND PUT THE IGNITORS IN CONTINUOUS. COULD NOT MAINTAIN VMC IN THE TOPS AND STARTED TO GET LIGHT PRECIP AND LIGHT TO MODERATE CHOP. ONLY A SMALL GREEN ECHO FINALLY SHOWED IN MAX GAIN AND JUST N OF THIS ECHO IS WHERE WE GOT THE PRECIP AND CHOP. CAPT (PF) SAID WE HAD A 'L ENG BLEED' EICAS AND THEN THAT THE L ENG HAD FLAMED OUT. I GAVE HIM THE DRIFTDOWN DATA FROM THE ENG OUT CRUISE PAGE AND COORDINATED A DSCNT WITH ATC. WITHIN 1-2 MINS OF FLAMING OUT; THE L ENG CAME BACK ON ITS OWN AND WE WERE NOW VMC AND ASKED TO LEVEL OFF AT FL310. CAPT THEN CONTACTED DISPATCH AND MAINT WHILE I FLEW AND HANDLED ATC. L ENG OP APPEARED NORMAL AND CAPT DECIDED; WITH MAINT DISPATCH AND MY INPUT; THAT IT WOULD BE SAFE TO CONTINUE. FLT CONTINUED AND WE LANDED SAFELY AT LAX WITHOUT FURTHER DIFFICULTIES. SUPPLEMENTAL INFO FROM ACN 518730: AFTER BRIEFING THE FLT ATTENDANTS; WE WERE INFORMED THAT THEY HAD HEARD A LOUD BANG PRIOR TO OUR DSCNT. AFTER CONSIDERATION AND EVALUATION; WE DETERMINED THAT THE BANG THEY HEARD WAS THE ENG LIGHTING OFF IN THE DSCNT DUE TO THE CONTINUOUS IGNITION. WE WERE DEBRIEFED BY MAINT AND 2 FAA OFFICIALS AT THE GATE IN LAX.