The symmetry in the structure of dynamical and adjoint symmetries of second-order differential equations

With each second-order differential equation 2 in the evolution space J1 (Mn+1) we associate, using a new differential operator AZ, four families of vector fields and 1-forms on J1 (Mn+1) providing a natural set-up for the study of symmetries, first integrals and the inverse problem for Z. We analyze the relations between the four families pointing out the symmetric structure of this set-up. When a Lagrangian for Z exists, the bijection between dynamical and dual symmetries is included in the whole context, suggesting the corresponding bijection between dual-adjoint and adjoint symmetries. As an application, we show how some results ofthe inverse problem can be framed naturally in this geometrical context.