Fast transients on power transmission systems, such as those produced by lightning and faults, are usually modelled by the Telegrapher's Equations. Because of corona, these equations are often nonlinear. It has long been recognised that the method of characteristics provides considerable advantages in dealing with these equations. To the best of the authors' knowledge, the first successful application of the eigenvector method of characteristics to nonlinear transients on lines is reported in this paper. In its present stage, the method described here can handle only single-phase lines with a static representation of corona. Nevertheless, this method lends itself to extensions that permit the handling of polyphasic lines, as well as the inclusion of dynamic corona models.