A transmission gearbox for a motor vehicle comprising: - a gearbox lever; - a ratchet actuated by means of said gearbox lever; - a selecting element actuated by said ratchet; - a stabilizer (114,115) associated to said selecting element, - wherein said selecting element has a star-like shape and comprises a plurality of valleys (112a, 112a') spaced out by a plurality of ridges, said valleys (112a, 112a') defining a corresponding gearbox gear; - a primary shaft apt to receive a driving torque (ω M); - a secondary shaft dragged by the primary shaft by means of a toothed coupling formed between first wheels (51, 50, 52) of the primary shaft and second wheels (6, 7, 80, 70, 60) of the secondary shaft, wherein said second wheels (6, 7, 80, 70, 60) of the secondary shaft comprise: ▪ at least an idle wheel (6, 60, 80) comprising a number of coupling slots; ▪ at least a coupling wheel (7, 70) comprising at least a number of coupling teeth apt to engage in said corresponding coupling slots; wherein the coupling wheel (7, 70) is apt to move axially on the secondary shaft when a rotation of said selecting element is triggered, for translating between a free configuration, wherein the coupling wheel (7, 70) is disengaged with respect to the idle wheel, and an engaged configuration wherein the coupling wheel (7, 70) is engaged with the idle wheel (6, 60, 80) by implementing a front coupling for transmitting the motion between the primary shaft and the secondary shaft, characterized in that said coupling slot comprises a portion of shallow recess and at least a first through-portion, wherein said portion of shallow recess has a depth (p) so that: - when said coupling tooth rests on said shallow recess, said selecting element performs a partial rotation (Qp), said partial rotation (Qp) being so as to move said stabilizer (114,115) between a first position wherein it lies at a first valley of the selecting element, and a second position wherein said stabilizer (114,115) climbs over the ridge adjacent to said first valley; - when said coupling tooth engages with said through-portion (161,161') said stabilizer (114,115) lies in a second valley of the selecting element, adjacent to said first valley. - a gearbox lever; - a ratchet actuated by means of said gearbox lever; - a selecting element actuated by said ratchet; - a stabilizer (114,115) associated to said selecting element, - wherein said selecting element has a star-like shape and comprises a plurality of valleys (112a, 112a') spaced out by a plurality of ridges, said valleys (112a, 112a') defining a corresponding gearbox gear; - a primary shaft apt to receive a driving torque (ω M); - a secondary shaft dragged by the primary shaft by means of a toothed coupling formed between first wheels (51, 50, 52) of the primary shaft and second wheels (6, 7, 80, 70, 60) of the secondary shaft, ▪ at least an idle wheel (6, 60, 80) comprising a number of coupling slots; ▪ at least a coupling wheel (7, 70) comprising at least a number of coupling teeth apt to engage in said corresponding coupling slots; - when said coupling tooth rests on said shallow recess, said selecting element performs a partial rotation (Qp), said partial rotation (Qp) being so as to move said stabilizer (114,115) between a first position wherein it lies at a first valley of the selecting element, and a second position wherein said stabilizer (114,115) climbs over the ridge adjacent to said first valley; - when said coupling tooth engages with said through-portion (161,161') said stabilizer (114,115) lies in a second valley of the selecting element, adjacent to said first valley.