Proper Weyl collineations in space-times

A Lorentzian manifold M is said to admit a Weyl collineation (WC) if there exists a vector field X along which the Lie derivative of the Weyl tensor, L_XC_bcd^a is zero. Historically the investigation of the WC symmetry is motivated for the role the Weyl tensor plays in algebraic classification of space-times according to their Petrov types. Recently some results have been published (see G. Shabbir, Ph.D. Thesis (2001) and G. S. Hall, Grav. Cosmol., 2 (1996) 270) in which it is shown that proper WC can only be admitted when the Petrov type of the given space-time is either N or O. Studying proper Weyl symmetry in space-times, we show that proper Weyl collineation form an infinite-dimensional vector space.