  An analysis of conformal geodesics in the Schwarzschild-de Sitter and
Schwarzschild-anti de Sitter families of spacetimes is given. For both families
of spacetimes we show that initial data on a spacelike hypersurface can be
given such that the congruence of conformal geodesics arising from this data
cover the whole maximal extension of canonical conformal representations of the
spacetimes without forming caustic points. For the Schwarzschild-de Sitter
family, the resulting congruence can be used to obtain global conformal
Gaussian systems of coordinates of the conformal representation. In the case of
the Schwarzschild-anti de Sitter family, the natural parameter of the curves
only covers a restricted time span so that these global conformal Gaussian
systems do not exist.
