What was found
AFATER LANDING FOLLOWING A LONG FLIGHT AT FL 430, CREW ATTEMPTED TO START APU FOR A QUICK TURN. APU FAILED TO START. THEREFORE, LEFT NR 2 ENGINE RUNNING FOR CROSSBLEED START. TROUBLESHOOTING REVEALED THAT FUEL SOLENOID VALVE FAILED TO OPEN WHEN COMMANDED BY ECU. IF ENGINES HAD FLAMED OUT, THIS EMERGENCY COMPONENT WOULD NOT HAVE STARTED. THEREFORE, CREW COULD NOT HAVE RESTARTED ENGINES. WHEN CREW RETURNED TO SBN, APU FUNCTIONAL NORMAL.