This equipment includes: bony anchoring members whereof at least one includes a proximal threaded stud for receiving a nut and a base portion for anchoring to a vertebra; one or two linking rods to be connected to the anchoring members and be attached to the vertebrae thereby parts for connecting the linking rod(s) to the anchoring members, and extension pieces for engaging on the proximal stud(s) of the anchoring member(s) for running down connecting parts on the stud(s) until they rest on the proximal zone(s) of the base portion(s) of the anchoring members. The stud of at least one anchoring member and the extension piece used therewith include positioning elements.