Abstract
The equations for a gravitational field in the presence of a two-parameter Abelian group of isometries (e.g. stationary axisymmetric fields or cylindrical waves) are shown to be equivalent to the construction of a connection (with vanishing curvature) on a principal bundle with an appropriate structure group. Examples of analogous constructions for other equations are mentioned. The gauge transformations leaving invariant the functional form of the connection transform solutions into solutions and provide a method for finding Bäcklund transformations, examples of which are given.
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