The null tetrad formalism is used to investigate the structure of the Einstein field equations for Kerr-Schild gravitational fields in the presence of an elastic solid source. It is shown that such equations may be reduced to five nonlinear-partial differential equations for five variables. It turns out that, when the interior solutions admit the same preferred null congruence of the vacuum ones and some compatibility conditions hold, it is possible to reduce them to a linear system and to develop a method of solution which closely;resembles the ''variation of the arbitrary constants'' for ordinary differential equations. In the present paper, the above technical framework is developed in general and applied to two simple examples, deferring to future work the approach to the Kerr-interior problem. (C) 1996 American Institute of Physics. [References: 18]
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