Symmetries of spacetime and their relation to initial value problems

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 types of gravity-matter systems preserves the symmetries of initial data specifications