Abstract

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.