In this series of papers, we study birational canonical transformations of the Painlevé system ℋ, that is, the Hamiltonian system associated with the Painlevé differential equations. We consider also τ -function related to ℋ and particular solutions of ℋ. The present article concerns the sixth Painlevé equation. By giving the explicit forms of the canonical transformations of ℋ associated with the affine transformations of the space of parameters of ℋ, we obtain the non-linear representation: G→G*, of the affine Weyl group of the exceptional root system of the type F4 A canonical transformation of G* can extend to the correspondence of the τ -functions related to ℋ. We show the certain sequence of τ -functions satisfies the equation of the Toda lattice. Solutions of ℋ, which can be written by the use of the hypergeometric functions, are studied in details.
Introduction..................................................269 • ̃ 1. Preliminaries and notations..............................271 • ̃ 2. The relations between i and kb............................277 • ̃ 3. The method T of constructing invariant eigendistributions...279 • ̃ 4. Proof of Lemma 3.5......................................290 • ̃ 5. A complete system of invariant eigendistributions...........306 • ̃ 6. Invariant eigendistributions of purely polynomial coefficients..309 • ̃ 7. The space _??_(ƒÉ) of invariant eigendistributions..............313 • ̃ 3. Construction of tempered invariant eigendistributions....... 321 • ̃ 9. Uniqueness of tempered invariant eigendistributions........330 References...................................................340
