Engine system and saddle-straddling type motor vehicle

A crankshaft (13) is rotated in a forward direction until a crank angle reaches a reverse rotation starting range after a stop and before a start-up of an engine (10), and the crankshaft (13) is rotated in a reverse direction until the crank angle goes over a start-up intake range from the reverse rotation starting range and reaches a start-up ignition range at the start-up of the engine (10). When the crank angle is in the start-up intake range, a fuel-air mixture made of fuel and air that is injected by an injector is introduced into a combustion chamber (31 a) from an intake passage (22) through an intake port (21). Further, when the crank angle is in the start-up ignition range, the fuel-air mixture in the combustion chamber (31 a) is ignited by an ignition plug.