The set of all such spaces is termed the null bundle of the space E. Necessary and sufficient conditions are given for selecting elements of the null bundle of E that have the same Ricci ten¬sor and/or the same Einstein tensor. In contrast to the general element of the null bundle, these elements are shown to be defined in terms of systems of partial differential equations that are homoge¬neous in the unknowns. It follows that all such elements of the null-bundle are continuously connected to the space E and lie along rays through E in the natural parameterization of the null bundle. These results provide the natural gener¬alization of the results of Kerr and Schild to the cases for which the space E is intrinsically curved and for which the momentum-energy tensor does not van¬ish.
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