Bifurcations of completely integrable 2-variable first-order partial differential equations

In this paper, we consider an implicit 2-variable first-order partial differential equation with complete integral. As an application of the Legendrian singularity theory, we give a generic classification of bifurcations of such differential equations with respect to the equivalence relation which is given by the group of point transformations following S. Lie's view. Since two one-parameter unfoldings of such differential equations are equivalent if and only if induced one-parameter unfoldings of integral diagrams are equivalent for generic equations, our normal forms are represented by one-parameter integral diagrams. (C) 2011 Elsevier Inc. All rights reserved.