Exact Solutions of a Nonlinear Klein-Gordon Equation in three-dimensional Space-time

We obtain a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations of second order and their solutions by a singularity analysis are classified. In particular, it has been shown that, whenever they have the Painlevé property, they can be transformed to standard forms by Moebius transformations of dependent variable and arbitrary smooth transformations of independent variable whose solutions, depending on the values of some parameters, are expressible in terms of either elementary functions or Jacobi elliptic functions.