Symmetries of the stationary axisymmetric vacuum Einstein equations which preserve asymptotic flatness

The stationary axially symmetric vacuum Einstein equations possess an extremely large group of symmetry transformations. The author gives a unified treatment of all of the currently known transformations, by considering their action on the boundary values of a general solution upon the symmetry axis. They use these results to construct two infinite families of transformations B(u) and D(u) which preserve asymptotic flatness. Attempting to understanding these transformations leads to the definition of a new gauge, the HKX gauge, in which they take their simplest form.