  The cycle-preserving symmetries for the nine two-dimensional real spaces of
constant curvature are collectively obtained within a Cayley-Klein framework.
This approach affords a unified and global study of the conformal structure of
the three classical Riemannian spaces as well as of the six relativistic and
non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both
Newton-Hooke and Galilean), and gives rise to general expressions holding
simultaneously for all of them. Their metric structure and cycles (lines with
constant geodesic curvature that include geodesics and circles) are explicitly
characterized. The corresponding cyclic (Mobius-like) Lie groups together with
the differential realizations of their algebras are then deduced; this
derivation is new and much simpler than the usual ones and applies to any
homogeneous space in the Cayley-Klein family, whether flat or curved and with
any signature. Laplace and wave-type differential equations with conformal
algebra symmetry are constructed. Furthermore, the conformal groups are
realized as matrix groups acting as globally defined linear transformations in
a four-dimensional "conformal ambient space", which in turn leads to an
explicit description of the "conformal completion" or compactification of the
nine spaces.
