  We study static spherically and hyperbolically symmetric solutions of the
Einstein equations in the presence of a conformally coupled scalar field and
compare them with those in the space filled with a minimally coupled scalar
field. We then study the Kantowski-Sachs cosmological solutions, which are
connected with the static solutions by the duality relations. The main
ingredient of these relations is an exchange of roles between the radial and
the temporal coordinates, combined with the exchange between the spherical and
hyperbolical two-dimensional geometries. A brief discussion of questions such
as the relation between the Jordan and the Einstein frames and the description
of the singularity crossing is also presented.
