Methods and systems for an electric drive axle of a vehicle are provided. A gear train in an electric drive axle system includes, in one example, an input shaft configured to rotationally couple to an electric motor-generator and including a first gear and an intermediate shaft including a second gear rotationally coupled to the first gear and a third gear and a fourth gear each configured to rotationally couple to a separate gear on an output shaft. In this example, the second gear, the third gear, and the fourth gear have different sizes, the third gear includes an axial extension having at outer surface, and an interior surface of the second gear circumferentially surrounds at least a portion of the outer surface of the axial extension.

A transmission including first and second input shafts rotatable about a first axis of rotation; at least one output shaft rotatable about at least one second axis of rotation, the at least one second axis of rotation non-collinear with the first axis of rotation; a plurality of lay shafts; a plurality of gears; and a plurality of half synchronizer clutches arranged to non-rotatably connect at least a portion of the plurality of gears to the plurality of lay shafts to transmit respective torque from the first and second input shafts to the at least one output shaft and disconnect the at least a portion of the plurality of gears from the plurality of lay shafts.