Painlevé equations through symmetry

The six Painleve equations (nonlinear ordinary differential equations of the second order with nonmovable singularities) have attracted the attention of mathematicians for more than a hundred years. These equations and their solutions (Painlev� 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 one of the aspects of the theory of Painlev� equations, namely to their symmetry properties. For several types of Painlev� equations (especially equations of types II and IV), the author studies families of transformations--the so-called B�cklund transformations--which transform solutions of a given Painlev� equations to solutions of the same equations 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 remarkable combinatorial structures of these symmetries and shows how they are related to the theory of \tau-functions associated to integrable systems.