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Shortly after Einstein published his general theory of relativity, thespherically symmetric solution of the vacuum field equations was discovered by Karl Schwarzschild, while Hermann Weyl showed that from any axisymmetric solution ψ of the Laplace equation∇²ψ = 0 (satisfying appropriate boundary conditions) the metric tensor of a static axisymmetric vacuum spacetime can be constructed.In particular, the Schwarzschild solution corresponds to a rather trivial solution of Laplace's equation expressed in terms of prolate spheroidal coordinates. It took about 45 years before Roy Kerr discovered what he calledthe 'rotating Schwarzschild solution', and an additional five years before I established that from any complex axisymmetric solution of the nonlinear equation (satisfying appropriate boundary conditions) the metric tensor of a stationary axisymmetric vacuum spacetime can be constructed. In particular,the Kerr solution corresponds to an extremely simple solution of thisequation expressed in terms of prolate spheroidal coordinates. Ever more complicated solutions of this equation (using prolate spheroidal coordinates) were discovered by Tomimatsu and Sato, but for none of the associated spacetimes has a reasonable material source been suggested.What the present book describes are some of the heroic efforts that have been undertaken to construct physically significant spacetimes by solvingthe vacuum Ernst equation. Unfortunately, thus far, no one has made muchprogress extending the Ernst equation approach to facilitate the investigationof spacetime within a stationary axisymmetric material source, where, for example, the stress–energy tensor is that of a perfect fluid. However, in 1994, Meinel and Neugebauer had the novel idea of focusing attention uponglobal solutions such as the spacetime geometry associated with a rotating infinitely thin disk of dust, where the material source can be represented by discontinuities in the metric tensor components.The first two chapters of this book are devoted to some basic ideas:in the introductory chapter 1 the authors discuss the concept of integrability, comparing the integrability of the vacuum Ernst equation with the integrability of nonlinear equations of Korteweg–de Vries (KdV) type,while in chapter 2 they describe various circumstances in which the vacuumErnst equation has been determined to be relevant, not only in connectionwith gravitation but also, for example, in the construction of solutionsof the self-dual Yang–Mills equations. It is also in this chapter thatone of several equivalent linear systems for the Ernst equation isdescribed.The next two chapters are devoted to Dmitry Korotkin's concept of algebro-geometric solutions of a linear system: in chapter 3 the structure of such solutions of the vacuum Ernst equation, which involve Riemann theta functions of hyperelliptic algebraic curves of any genus, is contrasted with the periodic structure of such solutions of the KdV equation. How such solutions can be obtained, for example, by solving a matrix Riemann–Hilbert problem and how the metric tensor of the associated spacetime can be evaluated is described in detail. In chapter 4 the asymptotic behaviour and the similarity structure of the general algebro-geometric solutions of the Ernst equation are described, and the relationship of such solutions to the perhaps more familiar multi-soliton solutions is discussed.The next three chapters are based upon the authors' own published research:in chapter 5 it is shown that a problem involving counter-rotatinginfinitely thin disks of matter can be solved in terms of genus two Riemann theta functions, while in chapter 6 the authors describe numerical methods that facilitate the construction of such solutions, and in chapter 7 three-dimensional graphs are displayed that depict all metrical fields of the associated spacetime.Finally, in chapter 8, the difficulties associated with extending the techniques espoused by these authors to other physically interesting problems (e.g., when electromagnetic fields are involved) are discussed.In addition to these eight chapters, there are two appendices. The firstconcerns the theory of Riemann surfaces and the second describes the relationship between the Ernst equation and twistor theory.It should be mentioned that this book may be considered to be an homage to Olaf Richter, whose very promising research life ended prematurely in November 2003. Indeed, a substantial part of the book is based upon his habilitation thesis. It is the reviewer's opinion that the resulting bookwill be more useful as a resource for those who are already well versed inthe subject of integrable systems than as an educational tool for novices who would like to enter this exciting branch of mathematical physics.