The Characterization of Third Order Ordinary Differential Equations Admitting a Transitive Fiber-Preserving Point Symmetry Group

Third order ordinary differential equations admitting a transitive symmetry group of fiber-preserving point transformations are completely characterized. We give the general expressions of these equations, as well as the canonical equations representing the equivalence classes. The symmetry groups admitted by these equations are also characterized. In some cases we give the transformation taking the canonical equation into the general one. The linearization problem for third order equations is solved.