CHARACTERIZATION OF THE ALGEBRAIC PROPERTIES OF FIRST INTEGRALS OF SCALAR ORDINARY DIFFERENTIAL-EQUATIONS OF MAXIMAL SYMMETRY

We undertake a study of the first integrals of linear nth order scalar ordinary differential equations with maximal symmetry. We establish p atterns for the first integrals associated with these equations. It is shown that second and third order equations are the pathological case s in the study of higher order differential equations. The equivalence of contact symmetries for third order equations to non-Cartan symmetr ies of second order equations is highlighted. (C) 1997 Academic Press.