An alternative approach to systems of second-order ordinary differential equations with maximal symmetry. Realizations of sl(n+2,R) by special functions

Using the general solution of the differential equation x¨(t)+g1(t)x˙+g2(t)x=0, a generic basis of the point-symmetry algebra sl(3,R) is constructed. Deriving the equation from a time-dependent Lagrangian, the basis elements corresponding to Noether symmetries are deduced. The generalized Lewis invariant is constructed explicitly using a linear combination of Noether symmetries. The procedure is generalized to the case of systems of second-order ordinary differential equations with maximal sl(n+2,R)-symmetry, and its possible adaptation to the inhomogeneous non-linear case illustrated by an example.