A cylinder, having a cylindrical body which is rotatable about a cylinder axis of rotation, is usable to process flat material. At least one group of a plurality of tools are distributed around the periphery of the cylinder body in a homogeneous manner. These tools are able to perform a working movement with respect to the cylindrical body and are coupled to a control device which effects that working movement. A first cam plate is traced by the control device of each tool group. A rotatable cover disk is also traced by the control device of each tool group. The cover plate is rotatable and is coupled, for its rotation, to the rotation of the cylinder. The cam plate has a peripheral section that controls the tool working movement, in cooperation with the control device. The control section covers a maximum of 1/n of the periphery of the cam plate, where n is a whole number integer equal to, or larger than 2, in addition to a section that does not control the working movement of the tool group and which constitutes the rest of the cam plate periphery. The cover disk includes a first group of sections of different radii, which are traced by the control device, and which control device also traces the sections of the cam disk during the rotations of the cylinder body. At least one of those cover disk sections has a first radius which enables cooperation between the control device and the cover disk section. At least one other of the sections has a second radius which blocks such cooperation. At least one second group of sectors is also provided on the cover disk and is arranged between two sections of the first group of sectors. The radii of the first sectors and the second sectors are not the same.