A limited slip differential (10) for a vehicle comprises a friction clutch (22) disposed between a differential gear mechanism (12) and a drive wheel axle (90), a clutch control cylinder (23) for coupling and decoupling the friction clutch (22), a normally closed type solenoid valve (27) for charging compressed air to the cylinder (23) from a pressure air source and a normally opened type solenoid valve (28) for discharging compressed air from the cylinder (23) to the air. The normally closed type and normally opened type solenoid valves (27, 28) are operated by a control unit (29) in response to a low µ road start, a low µ road run, a straight run and a brake so as to charge and discharge compressed air to and from the cylinder (23) and thereby to control air pressure in the cylinder (23). A power device (60) for a vehicle is composed by adding an engine control cylinder (65) disposed in an acceleration control mechanism (64) and an engine control valve (66) for charging and discharging compressed air to the cylinder, and wherein during a low µ road run of the vehicle the engine control valve (66) is operated by the control unit (29) to charge compressed air to the engine control cylinder (65) thereby to cause the engine control cylinder (65) an acceleration return.