  The linearization problem by use of the Cartan equivalence method for scalar
third-order ODEs via point transformations was solved partially in [1,2]. In
order to solve this problem completely, the Cartan equivalence method is
applied to provide an invariant characterization of the linearizable
third-order ordinary differential equation u'''=f(x,u,u',u'') which admits a
four-dimensional point symmetry Lie algebra. The invariant characterization is
given in terms of the function f in a compact form. A simple procedure to
construct the equivalent canonical form by use of an obtained invariant is also
presented. The method provides auxiliary functions which can be utilized to
efficiently determine the point transformation that does the reduction to the
equivalent canonical form. Furthermore, illustrations to the main theorem and
applications are given.
