The guidance system comprises two guidance devices (16) mounted in opposite directions either side of the axle (2) and each comprising: a guidance arm (17) terminated by at least one guidance roller (18); a set of articulated connecting rods serving as a deformable link interface (23) between the axle and the guidance arm; and a jack (29) with two independent and opposing rods (30, 31), having a “rods extended” position in which its two rods lock the set of connecting rods to render it rigid and orient the axle, and a “rods retracted” position in which the set of connecting rods is free and allows a lateral oscillation. Preferably, the jack (29) offers an additional function for damping the lateral oscillation. This invention is of interest in the field of the guidance of public transport road vehicles by a rail.