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A scheme is introduced which yields, beginning with any source-free solution of Einstein's equation with two commuting Killing fields for which a certain pair of constants vanish (e.g., the exterior field of a rotating star), a family of new exact solutions. To obtain a new solution, one must specify an arbitrary curve (up to parametrization) in a certain three-dimensional vector space. Thus, a single solution of Einstein's equation generates a family of new solutions involving two arbitrary functions of one variable. These transformations on exact solutions form a non-Abelian group. The extensive algebraic structure thereby induced on such solutions is studied.
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Robert Geroch (1972) studied this question.
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