AN LJ25 IN CRUISE FLT AT FL410 EXPERIENCED A FLAMEOUT OF L ENG. CREW DECLARED EMER AND DSNDED FOR A SUCCESSFUL RELIGHT. AFTER LNDG AND STATIC RUN-UP; THEY LATER HAD ANOTHER FLAMEOUT OF THE L ENG; AGAIN AT FL410.

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AT FL410 WHILE IN A L TURN; WE HAD A FLAMEOUT ON THE L ENG ON A LEAR 25. THEREFORE; WE COULD NOT HOLD ALT. WE NOTIFIED CTR 2 TIMES THAT WE HAD A FLAMEOUT AND NEEDED TO DSND. THEY WOULD NOT LET US DSND DUE TO TFC. THEN WE DECLARED AN EMER. THEY GAVE US A R TURN TO 270 DEGS AT FL350. THEN WE TOLD THEM WE NEEDED FL250; TO ATTEMPT A RELIGHT AND GO TO ZZZ1. THEY THEN CLRED US TO FL250. AT FL250 WE GOT A RELIGHT. THEY ASKED US IF WE WOULD LIKE TO CONTINUE THE EMER. WE TOLD THEM NO; THAT WE HAD BOTH ENGS BACK AND EVERYTHING WAS NORMAL. THEY GAVE US A R TURN DIRECT TO ZZZ1. ON THE GND I CALLED OUR HEAD MECH. HE HAD ME DO A STATIC RUN-UP ACCORDING TO HIS SPECS. EVERYTHING CHKED OUT NORMAL. HE CLRED US TO FLY. WE THEN WENT BACK UP. AT ABOUT XA30 AT FL410 WHILE IN ANOTHER L TURN; WE HAD ANOTHER FLAMEOUT OF THE L ENG. WE STARTED TO LOSE ALT. WE NOTIFIED ATC WHO CLRED US TO FL350. WE TOLD THEM WE WANTED TO GO TO ZZZ2 AND FL250 TO ATTEMPT A RELIGHT. THEY CLRED US DIRECT AND FL250. AT FL250 WE GOT THE RELIGHT. WE CONTINUED. THE NEXT DAY I TALKED TO OUR HEAD MECH AGAIN AND DECIDED THE BEST THING TO DO WOULD BE TO FLY HOME AT FL250 WHICH IS IN THE RELIGHT ENVELOPE; SO THAT MAINT COULD DO A MORE THOROUGH CHK.