at one end by the presence of the plate 14 on its portion 32 and at its other end by the presence of the projections having surfaces 28 at the ends of the tongues 30 since each of the surfaces 28 is suitable for coming into contact with the surface 56 of the orifice 18 secondly when the plate 14 comes into contact with the rear face 60 of a half shell the surface 28 of each of the tongues 30 is spaced apart from the surface 56 of the half shell conversely when each of the surfaces 28 of each of the tongues 30 is in contact with the surface 56 of the half shell then the plate 14 is no longer in contact with the surface 60 of the half shell finally in use the elongate member 2 is then inserted into the assembly the operation of the position adjustment system 1 is described below with reference to fig5 and 7 a spinal column has a vertebra v 3 which presents multiple fractures this vertebra v 3 is associated with two healthy vertebrae v 1 and v 2 on either side via vertebral disks i 1 and i 2 as it is known to those of ordinary skill in the art it is assumed that the elongate member 2 is connected to an element e 1 of a surgical instrument that is engaged in the vertebra vi while the body.