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Rational surfaces associated with affine root systems and geometry of the Painlevé equations. (English) Zbl 1010.34083
Here, the author provides an understanding of (discrete and continuous) Painlevé equations from a purely geometric point of view. The paper is motivated by previous work of Okamoto (and later Takano), but turns this work on its head by beginning with geometric structures and then deriving equations, along with many of their important properties. Thus, in addition to Painlevé equations, this geometric framework is also shown to yield Bäcklund transformations, special integrals and elliptic function limits. The author also gives a generic discrete equation which contains all other discrete equations which appear in the paper.

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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