  We consider covariant metric theories of coupled gravity-matter systems
satisfying the following two conditions: First, it is assumed that, by a
hyperbolic reduction process, a system of first order symmetric hyperbolic
partial differential equations can be deduced from the matter field equations.
Second, gravity is supposed to be coupled to the matter fields by requiring
that the Ricci tensor is a smooth function of the basic matter field variables
and the metric. It is shown then that the ``time'' evolution of these type of
gravity-matter systems preserves the symmetries of initial data specifications.
