A transmission gearbox for a motor vehicle (M) comprising: - a gearbox lever ; - a ratchet actuated by means of said gearbox lever ; - a selecting element actuated by said ratchet ; - a stabilizer associated to said selecting element , - wherein said selecting element has a star-like shape and comprises a plurality of valleys spaced out by a plurality of ridges , said valleys 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 of the primary shaft and second wheels of the secondary shaft , wherein said second wheels of the secondary shaft comprise: ▪ at least an idle wheel comprising a number of coupling slots ; ▪ at least a coupling wheel comprising at least a number of coupling teeth apt to engage in said corresponding coupling slots ; wherein the coupling wheel is apt to move axially on the secondary shaft when a rotation of said selecting element is triggered, for translating between a free configuration (A), wherein the coupling wheel is disengaged with respect to the idle wheel , and an engaged configuration (B) wherein the coupling wheel is engaged with the idle wheel 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 between a first position wherein it lies at a first valley of the selecting element , and a second position wherein said stabilizer climbs over the ridge adjacent to said first valley ; - when said coupling tooth engages with said through-portion said stabilizer 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 associated to said selecting element , - wherein said selecting element has a star-like shape and comprises a plurality of valleys spaced out by a plurality of ridges , said valleys 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 of the primary shaft and second wheels of the secondary shaft , ▪ at least an idle wheel comprising a number of coupling slots ; ▪ at least a coupling wheel 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 between a first position wherein it lies at a first valley of the selecting element , and a second position wherein said stabilizer climbs over the ridge adjacent to said first valley ; - when said coupling tooth engages with said through-portion said stabilizer lies in a second valley of the selecting element , adjacent to said first valley . <EOS>