Relationships between the group-theoretic and soliton-theoretic techniques for generating stationary axisymmetric gravitational solutions

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Abstract
We investigate the precise interrelationships between several recently developed solution‐generating techniques capable of generating asymptotically flat gravitational solutions with arbitrary multipole parameters. The transformations we study in detail here are the Lie groups Q and Q̃ of Cosgrove, the Hoenselaers–Kinnersley–Xanthopoulos (HKX) transformations and their SL(2) tensor generalizations, the Neugebauer–Kramer discrete mapping, the Neugebauer Bäcklund transformations I1 and I2, the Harrison Bäcklund transformation, and the Belinsky–Zakharov (BZ) one‐ and two‐soliton transformations. Two particular results, among many reported here, are that the BZ soliton transformations are essentially equivalent to Harrison transformations and that the generalized HKX transformation may be deduced as a confluent double soliton transformation. Explicit algebraic expressions are given for the transforms of the Kinnersley–Chitre generating functions under all of the above transformations. In less detail, we also study the Kinnersley–Chitre β transformations, the non‐null HKX transformations, and the Hilbert problems proposed independently by Belinsky and Zakharov, and Hauser and Ernst. In conclusion, we describe the nature of the exact solutions constructible in a finite number of steps with the available methods.