Primary classification of symmetries from the solution manifold in Classical Mechanics

The symmetries of the equations of motion of a classical system are characterized in terms of vector field subalgebras of the whole diffeomorphism algebra of the solution manifold (the space of initial constants endowed with a symplectic structure). Among them, naturally arises the subalgebra of Hamiltonian (contact) vector fields corresponding to (jet-prolongued) point symmetries, those not corresponding to point symmetries and the remaining symmetries being associated with non-Hamiltonian (hence non-symplectic) non-strict contact symmetries.