In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

Abstract. The fourth Painlevé equation PIV is known to have symmetry of the affine Weyl group of type A (1) 2 with respect to the Bäcklund transformations. We introduce a new representation of PIV , called the symmetric form, by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of PIV is given in terms of this representation. Through the symmetric form, it turns out that PIV is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions PIV , called Okamoto polynomials, are expressible in terms of the 3-reduced Schur functions.It is known by K. Okamoto [7] that the fourth Painlevé equation has symmetries under the affine Weyl group of type A 2 -symmetries become clearly visible. By means of this representation, we clarify the internal relation between the fourth Painlevé equation and the modified KP hierarchy. We obtain in particular a complete description of the rational solutions of the fourth Painlevé equation in terms of Schur functions. This implies that the so-called Okamoto polynomials, which arise from the τ -functions for rational solutions, are in fact expressible by the 3-reduced Schur functions. 1 §1. A symmetric form of the fourth Painlevé equationThe fourth Painlevé equation P IV is the following second order ordinary

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A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painlevé type are discussed.