“The Painlevé equations themselves are really a wonder. They still continue to give us fresh mysteries ... One reason that I wrote this book is to tell you how impressed I am by the mysteries of the Painlevé equations.”

—from the Preface

The six Painlevé equations (nonlinear ordinary differential equations of the second order with nonmovable singularities) have attracted the attention of mathematicians for more than 100 years. These equations and their solutions, the Painlevé transcendents, nowadays play an important role in many areas of mathematics, such as the theory of special functions, the theory of integrable systems, differential geometry, and mathematical aspects of quantum field theory.

The present book is devoted to the symmetry of Painlevé equations (especially those of types II and IV). The author studies families of transformations for several types of Painlevé equations—the so-called Bäcklund transformations—which transform solutions of a given Painlevé equation to solutions of the same equation with a different set of parameters. It turns out that these symmetries can be interpreted in terms of root systems associated to affine Weyl groups. The author describes the remarkable combinatorial structures of these symmetries, and shows how they are related to the theory of \(\tau\)-functions associated to integrable systems.

Prerequisites include undergraduate calculus and linear algebra with some knowledge of group theory. The book is suitable for graduate students and research mathematicians interested in special functions and the theory of integrable systems.