  We adopt a reference-metric approach to generalize a covariant and conformal
version of the Z4 system of the Einstein equations. We refer to the resulting
system as ``fully covariant and conformal", or fCCZ4 for short, since it is
well suited for curvilinear as well as Cartesian coordinates. We implement this
fCCZ4 formalism in spherical polar coordinates under the assumption of
spherical symmetry using a partially-implicit Runge-Kutta (PIRK) method and
show that our code can evolve both vacuum and non-vacuum spacetimes without
encountering instabilities. Our method does not require regularization of the
equations to handle coordinate singularities, nor does it depend on
constraint-preserving outer boundary conditions, nor does it need any
modifications of the equations for evolutions of black holes. We perform
several tests and compare the performance of the fCCZ4 system, for different
choices of certain free parameters, with that of BSSN. Confirming earlier
results we find that, for an optimal choice of these parameters, and for
neutron-star spacetimes, the violations of the Hamiltonian constraint can be
between 1 and 3 orders of magnitude smaller in the fCCZ4 system than in the
BSSN formulation. For black-hole spacetimes, on the other hand, any advantages
of fCCZ4 over BSSN are less evident.
