New Bäcklund Transformations and Superposition Principle for Gravitational Fields with Symmetries

Vector Bäcklund transformations which relate solutions of the vacuum Einstein equations having two commuting Killing fields are introduced. Such transformations generalize those found by Pohlmeyer in connection with the nonlinear $\sigma$ model. A simple algebraic superposition principle, which permits the combination of Bäcklund transforms in order to get new solutions, is given. The superposition preserves the asymptotic flatness condition, and the whole scheme is manisfestly O(2,1) invariant.