In this talk, we give the necessary and sufficient condition, that a homogeneous partial differential equation with two variables can be reduced to a homogeneous ordinary one under a certain change of variables, by means of the commutator with a first order partial differential operator which characterizes a homogeneity. Moreover we obtain the explicit representation of the ordinary differential equation.
Under the reduction strategy guaranteed by the necessary and sufficient condition, we characterize such partial differential equations and investigate the succession of its singularities after the reduction. This fact enable us to understand that local structures of the solutions to partial differential equations on the universal covering space can be described by global structures of those to ordinary ones.
This work is joint work with Takuya WATANABE.