Final answer:
The gear case is the component of a Bendix dual magneto that is shared by both ignition systems.
Step-by-step explanation:
The component of a Bendix dual magneto that is shared by both ignition systems is the gear case. The gear case houses the rotating magnet rotor, which generates the electrical current for both the left and right ignition systems. This allows for redundancy and increased reliability in the event of a failure in one of the ignition systems.