A method is described for constructing, from any source-free solution of Einstein's equations which possesses a Killing vector, a one-parameter family of new solutions. The group properties of this transformation are discussed. A new formalism is given for treating space-times having a Killing vector.

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It is shown that, given any set of initial data for Einstein's equations which satisfy the constraint conditions, there exists a development of that data which is maximal in the sense that it is an extension of every other development. These maximal developments form a well-defined class of solutions of Einstein's equations. Any solution of Einstein's equations which has a Cauchy surface may be embedded in exactly one such maximal development