The presence of two Killing symmetries yields a considerable simplification of the field equations. In
fact, for certain matter models the latter become completely integrable [127], provided that
the Killing fields satisfy the Frobenius conditions. Space-times admitting two Killing fields
provide the framework for both the theory of colliding gravitational waves and the theory of
rotating black holes [37]. Although dealing with different physical subjects, the theories are
mathematically closely related. As a consequence of this, various stationary and axisymmetric solutions
which have no physical relevance give rise to interesting counterparts in the theory of colliding
waves.^{56}

This section reviews the structure of the stationary and axisymmetric field equations. We start by recalling the circularity problem (see also Section 2.1 and Section 3.5). It is argued that circularity is not a generic property of asymptotically flat, stationary and axisymmetric space-times. If, however, the symmetry conditions for the matter fields do imply circularity, then the reduction with respect to the second Killing field simplifies the field equations drastically. The systematic derivation of the Kerr–Newman metric and the proof of its uniqueness provide impressive illustrations of this fact.

6.1 Integrability properties of Killing fields

6.2 Boundary value formulation

6.3 The Ernst equations

6.4 The uniqueness theorem for the Kerr–Newman solution

6.2 Boundary value formulation

6.3 The Ernst equations

6.4 The uniqueness theorem for the Kerr–Newman solution

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