The invention relates to a head-up display system embedded in an aircraft pilotable by a pilot and having a head-down display screen. The display system includes: a head-up display screen configured to be arranged in the cockpit in a Plurality of positions and to be observed by the pilot from an observation point along a viewing field; at least one position, called interposition position, of the head-up screen corresponding to a position in which a part, called interposed part, is arranged across from a part of the head-down display screen in the viewing field between the observation point and this part; a display control module configured to display a conformal display area and when the head-up display screen is in the interposition position, a non-conformal display area in the interposed part.