Symmetries of differential equations. II

The importance of the non‐pointlike transformations of symmetry is vindicated in relation with the first integrals. A new first integral of a broad class of systems of second order differential equations is obtained out of a symmetry of them without having to impose the restriction that the system be equivalent to a Lagrangian system. The existence of a reciprocal relationship among the local infinitesimal symmetries (l.i.s.) of a Newtonian system of differential equations and the pseudosymmetries of the associated dynamical system is proved. Several applications are developed, and some open problems concerning the dynamical systems of constant divergence are proposed.