A LEAR25 IN CLB AT FL370 HAD THE R ENG FLAME OUT. DSNDED AND SUCCESFULLY RESTARTED ENG. CLBING THROUGH FL400; THE R ENG FLAMED OUT AGAIN.

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WE DEPARTED ZZZ; TO RETURN TO ZZZ1 UNDER PART 91. WE ARE A PART 135 OPERATOR. BUT THE FLT IN QUESTION WAS AN EMPTY NON REVENUE LEG HOME TO ZZZ1. DURING THE CLB TO CRUISE ALT; WE ENCOUNTERED CLEAR AIR TURB AT FL370 AND THE R ENG FLAMED OUT. WE THEN INFORMED ATC OF THE SIT. WE DID NOT DECLARE AN EMER. WE DSNDED TO THE RELIGHT ENVELOPE AND RESTARTED THE R ENG. THEN BEGAN A NEW CLB TO OUR FILED ALT OF FL410. AT FL400; THE R ENG FLAMED AGAIN. WE ADVISED ATC OF THE FAILURE. ADVISED THAT WE NEEDED TO DSND TO THE RELIGHT ENVELOPE TO RELIGHT. AGAIN; WE DID NOT DECLARE AN EMER. ATC ASKED IF WE NEEDED ASSISTANCE; AND WE SAID NO. WE RESTARTED THE ENG AND CLB TO FL330 AND CHANGED DEST TO ZZZ2. WE LANDED AND PURCHASED FUEL. I ADVISED MY EMPLOYER OF THE SIT. WE THEN FLEW FROM ZZZ2 TO ZZZ1 AT 17500 FT; WELL WITHIN THE RELIGHT ENVELOPE. UPON ARR; I GNDED THE ACFT PER OUR OP SPECS. AT THE TIME OF THIS WRITING; I'M WAITING FOR MAINT'S ANALYSIS OF THE ENG. I WAS FLYING PLT ON BOTH ZZZ TO ZZZ2 OCCURRENCES. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR STATED MAINT MADE SEVERAL CHKS AND THEN REPLACED THE ENG FUEL CTL. ON TEST FLT THE ENG PERFORMED OK. THE RPTR IS UNCERTAIN IF THE CLEAR AIR TURB ACTUALLY CAUSED THE FIRST FLAME OUT.