Conformal symmetries and fixed points in space-time

A general study is made of conformal vector fields on a four-dimensional Lorentz manifold with particular emphasis being laid on the structure of the zeros (critical points) of such vectors fields. The implications for general relativity are investigated and a discussion of conformal vector fields in generalized plane wave space-times is given. An attempt is made to clarify the well-known theorem of Bilyalov and Defrise-Carter.