  The general-covariant Z4 formalism is further analyzed. The gauge conditions
are generalized with a view to Numerical Relativity applications and the
conditions for obtaining strongly hyperbolic evolution systems are given both
at the first and the second order levels. A symmetry-breaking mechanism is
proposed that allows one, when applied in a partial way, to recover previously
proposed strongly hyperbolic formalisms, like the BSSN and the Bona-Mass\'o
ones. When applied in its full form, the symmetry breaking mechanism allows one
to recover the full five-parameter family of first order KST systems. Numerical
codes based in the proposed formalisms are tested. A robust stability test is
provided by evolving random noise data around Minkowski space-time. A strong
field test is provided by the collapse of a periodic background of plane
gravitational waves, as described by the Gowdy metric.
