A clamp fastener for fastening an accessory detachably and adjustably by position to a suitable fastening support, said clamp fastener comprising - a shaft that is supported to the fastening support which is arranged to serve as the frame of said clamp fastener; - a clamp including two adjacent plier elements, a first and a second plier element (41, 42), a gap with a diameter (b) between the plier elements, said gap defining the width of the opening in the rest position of said clamp, said clamp further containing a first spring element; in which gap there is arranged a channel for the support bar of an accessory in order to fasten said accessory to the clamp and to the clamp fastener; - first and second gear ring elements (6; 61, 62) arranged in a mutually matching position; - a second spring element (7; 71) between the gear rings (6; 61, 62); and - a clamp actuator; said clamp, gear ring elements (6; 61, 62) and the second spring element (7; 71) being fitted in an intermediate space of the fastening support and interconnected by a shaft passing at right angles through their center region; and at the first end of said shaft, outside the fastening support, there is attached a clamp actuator for pressing the plier elements (41, 42) of the clamp together, and respectively for opening them in order to fasten the support bar in the channel and respectively for detaching it from the channel; the shaft is in the region of the clamp provided with two shoulders (33, 34) that are mutually spaced apart at a distance (a), the first of said shoulders being arranged outside the clamp, towards the outer surface (41 b) of the first plier element, and the second being arranged inside the clamp, towards the inner surface of the second plier element, and that in between said second plier element and the fastening support, there are arranged mutually facing gear ring elements (6; 61, 62), and a second spring element (7; 71), wherein the distance of the shoulders (33, 34) is shorter than the added length of the diameter (b) (in the rest position of the clamp) of the gap and the thickness (c) of the first plier element, i.e. a < b + c; characterized in that - the distance of shoulders (a) is bigger than the thickness (c) of the first plier element, - the clamp actuator is an eccentric lever wherein the eccentric lever, especially the eccentric lever head, is provided with adjusting elements for adjusting the position of the first end of the shaft in relation to the eccentric lever head, wherein the adjusting elements comprise a threaded part arranged at the first end of the shaft, and a cross nut arranged at the head of the eccentric lever, said cross nut being provided with an aperture for the shaft, said threadings of the threaded part and of the shaft aperture being mutually compatible; and a stopper element that is fitted in a hole, at right angles to the axis aperture of the cross nut, said stopper element and hole being provided with mutually compatible threadings, said stopper element being arranged, by rotating it in one direction in the hole, to be pressed against the first shaft end placed in the aperture of the cross nut, particularly against the threaded part, and to lock the shaft in place with respect to the eccentric lever head, and respectively, by rotating the stopper element in the opposite direction, to release the first end of the shaft, so that the shaft and the eccentric lever can be mutually rotated in the threadings of the threaded part and the shaft aperture and adjusted in position. - a shaft that is supported to the fastening support which is arranged to serve as the frame of said clamp fastener; - a clamp including two adjacent plier elements, a first and a second plier element (41, 42), a gap with a diameter (b) between the plier elements, said gap defining the width of the opening in the rest position of said clamp, said clamp further containing a first spring element; in which gap there is arranged a channel for the support bar of an accessory in order to fasten said accessory to the clamp and to the clamp fastener; - first and second gear ring elements (6; 61, 62) arranged in a mutually matching position; - a second spring element (7; 71) between the gear rings (6; 61, 62); and - a clamp actuator; - the distance of shoulders (a) is bigger than the thickness (c) of the first plier element, - the clamp actuator is an eccentric lever wherein the eccentric lever, especially the eccentric lever head, is provided with adjusting elements for adjusting the position of the first end of the shaft in relation to the eccentric lever head, wherein the adjusting elements comprise a threaded part arranged at the first end of the shaft, and a cross nut arranged at the head of the eccentric lever, said cross nut being provided with an aperture for the shaft, said threadings of the threaded part and of the shaft aperture being mutually compatible; and a stopper element that is fitted in a hole, at right angles to the axis aperture of the cross nut, said stopper element and hole being provided with mutually compatible threadings, said stopper element being arranged, by rotating it in one direction in the hole, to be pressed against the first shaft end placed in the aperture of the cross nut, particularly against the threaded part, and to lock the shaft in place with respect to the eccentric lever head, and respectively, by rotating the stopper element in the opposite direction, to release the first end of the shaft, so that the shaft and the eccentric lever can be mutually rotated in the threadings of the threaded part and the shaft aperture and adjusted in position.