  A general covariant extension of Einstein\'{}s field equations is considered
with a view to Numerical Relativity applications. The basic variables are taken
to be the metric tensor and an additional four-vector $Z_\mu$. Einstein's
solutions are recovered when the additional four-vector vanishes, so that the
energy and momentum constraints amount to the covariant algebraic condition
$Z_\mu=0$. The extended field equations can be supplemented by suitable
coordinate conditions in order to provide symmetric hyperbolic evolution
systems: this is actually the case for either harmonic coordinates or normal
coordinates with harmonic slicing.
