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29Twisting vacuum solutionsIn the preceding chapter, we treated the non-twisting solutions (of theRobinson–Trautman class) in some detail, giving or indicating nearly allproofs, and showing how the variety of known special solutions fits intothe framework of the canonical form of the metric and field equations.It is impossible to present the solutions with twisting degenerate eigenraysin this detailed manner, because it would nearly fill an extra volume.We share this problem with most of the authors writing on the subject.The necessity of presenting the complicated calculations in a compressedform makes some of the papers almost unreadable, and it is sometimes aformidable task merely to check the calculations. What we will do hereand in the following chapters is to show why, how and how far the integrationprocedure of the field equations works, and what classes of solutionsare known.29.1 Twisting vacuum solutions – the field equations29.1.1 The structure of the field equationsTo get a better understanding of the structure of the vacuum field equations,we follow Sachs (1962) in dividing them into three sets:six main equations: R 11 = R 12 = R 14 = R 44 = 0 (29.1)one trivial equation: R 34 = 0 (29.2)three supplementary conditions: R 13 = R 33 = 0 (29.3)(the indices refer to the tetrad (3.8)). The reason for this splitting is thefollowing property (which can be proved by application of the Bianchiidentities): if k is a geodesic and diverging (k a ;a ̸= 0) null congruence,437