Contents

Contentsxvii34.1.3 Complex invariance transformations 52534.1.4 Stationary axisymmetric vacuum fields 52634.2 Prolongation structure for the Ernst equation 52934.3 The linearized equations, the Kinnersley–Chitre B group andthe Hoenselaers–Kinnersley–Xanthopoulos transformations 53234.3.1 The field equations 53234.3.2 Infinitesimal transformations and transformationspreserving Minkowski space 53434.3.3 The Hoenselaers–Kinnersley–Xanthopoulos transformation53534.4 Bäcklund transformations 53834.5 The Belinski–Zakharov technique 54334.6 The Riemann–Hilbert problem 54734.6.1 Some general remarks 54734.6.2 The Neugebauer–Meinel rotating disc solution 54834.7 Other approaches 54934.8 Einstein–Maxwell fields 55034.9 The case of two space-like Killing vectors 55035 Special vector and tensor fields 55335.1 Space-times that admit constant vector and tensor fields 55335.1.1 Constant vector fields 55335.1.2 Constant tensor fields 55435.2 Complex recurrent, conformally recurrent, recurrent andsymmetric spaces 55635.2.1 The definitions 55635.2.2 Space-times of Petrov type D 55735.2.3 Space-times of type N 55735.2.4 Space-times of type O 55835.3 Killing tensors of order two and Killing–Yano tensors 55935.3.1 The basic definitions 55935.3.2 First integrals, separability and Killing or Killing–Yano tensors 56035.3.3 Theorems on Killing and Killing–Yano tensors in fourdimensionalspace-times 56135.4 Collineations and conformal motions 56435.4.1 The basic definitions 56435.4.2 Proper curvature collineations 56535.4.3 General theorems on conformal motions 56535.4.4 Non-conformally flat solutions admitting properconformal motions 567