As reported to the FAA. Reports are unverified, and codes are left as the FAA publishes them.
What was found
ROUTE JVL TO GKY ATTEMPTED TO CLIMB OVER WEATHER FL430 TO FL450 TO TOP THUNDERSTORM RT ENGINE FLAME-OUT. DESCENDED TO FL250 PERFORMED AIR START, ALL OPERATION NORMAL. CONTINUED TO FINAL DESTINATION ADS. MAINTENANCE TO CK ENGINE WILL UPDATE REPORT. MAINTENANCE COMPLIED WITH FLAME-OUT INSPECTION IN ACCORDANCE WITH GE SEI-186, NO DEFECTS NOTED AT THIS TIME. AIRCRAFT RETURNED TO SERVICE.