A motor vehicle including a structure with: - two structural profiled sections (108; 208; 308; 408; 508) extending longitudinally generally parallel to one another; and - a shock absorbing device (102; 202; 302; 402; 502), including: o two profiled sections which are deformable by compression (106; 206; 306; 406; 506), disposed parallel to one another and intended to be fixed in the extension of the two longitudinal structural profiled sections (108; 208, 308, 408, 508) of the vehicle, respectively; ∘ a transverse beam (104; 204; 304; 404; 504) fixed to the front ends of the deformable profiled sections (106; 206; 306; 406; 506), the beam having a profile that is generally curved towards the front; and ∘ means for controlled deformation (130, 230; 330; 430; 530) disposed on the transverse beam (104; 24; 304; 404; 504) and configured to act on at least one of the deformable profiled sections (106; 206; 306; 406; 506) during a straightening deformation of said beam, so as to exert a bending torque on the structural profiled section(s), characterized in that the means for controlled deformation (130; 230; 330; 430; 530) of said shock absorbing device (102; 202; 302; 402; 502) are disposed exclusively on the right-hand side of the vehicle and form an arm projecting from the rear face of the transverse beam, configured to cooperate by engagement with an attachment plate of the deformable profiled section(s) to the structural profiled section(s) of the vehicle during the straightening deformation of said beam, and to override said plate during a compression deformation of the corresponding deformable profiled section without straightening deformation of said beam. - two structural profiled sections (108; 208; 308; 408; 508) extending longitudinally generally parallel to one another; and - a shock absorbing device (102; 202; 302; 402; 502), including: o two profiled sections which are deformable by compression (106; 206; 306; 406; 506), disposed parallel to one another and intended to be fixed in the extension of the two longitudinal structural profiled sections (108; 208, 308, 408, 508) of the vehicle, respectively; ∘ a transverse beam (104; 204; 304; 404; 504) fixed to the front ends of the deformable profiled sections (106; 206; 306; 406; 506), the beam having a profile that is generally curved towards the front; and ∘ means for controlled deformation (130, 230; 330; 430; 530) disposed on the transverse beam (104; 24; 304; 404; 504) and configured to act on at least one of the deformable profiled sections (106; 206; 306; 406; 506) during a straightening deformation of said beam, so as to exert a bending torque on the structural profiled section(s), o two profiled sections which are deformable by compression (106; 206; 306; 406; 506), disposed parallel to one another and intended to be fixed in the extension of the two longitudinal structural profiled sections (108; 208, 308, 408, 508) of the vehicle, respectively; ∘ a transverse beam (104; 204; 304; 404; 504) fixed to the front ends of the deformable profiled sections (106; 206; 306; 406; 506), the beam having a profile that is generally curved towards the front; and ∘ means for controlled deformation (130, 230; 330; 430; 530) disposed on the transverse beam (104; 24; 304; 404; 504) and configured to act on at least one of the deformable profiled sections (106; 206; 306; 406; 506) during a straightening deformation of said beam, so as to exert a bending torque on the structural profiled section(s),