Abstract: A machining unit for the production of out-of-center/convex geometries on turned parts using an NC turning machine with at least one horizontal and one vertical machine axis comprises a highly-dynamic, controllable, integral three-phase servomotor having a shaft forming an additional tool axis, as a programmable direct drive for the production of out-of-center/convex geometries on current parts. The additional tool axis is capable of being NC controlled. The shaft has a connecting arm which carries one or more cutting tools for machining the parts. Spaced hydrostatic radial bearings and a hydrostatic axial bearing mount the shaft. A rotational measuring system measures rotation of the shaft. The shaft rotational measuring system, hydrostatic axial bearing and hydrostatic radial bearings as well as the servomotor are mounted in a common housing, with the connecting arm with cutting tools projecting from one side of the housing.