Fuel cut-off valve
It usually dies either when you (try to) start the engine or when you stop it, rarely while it's running. The engine will not start because it will not be getting any fuel. The reason is usually overheating and deformations in the coil whichresult in the valve plunger getting stuck, or internal shorts or open circuits (due to heat) in the coil.
It is simple to check for--when you turn the ignition on, the valve makes a faint click. If you cannot hear it through the components blocking your way, use the stethoscopic method to listen to it. If you cannot hear the click but you can measure the voltage on the contact on top of the valve, it is positively dead. Replacing is straightforward.