Folding transformations of the Painlevé equations
From authors' abstract: ``New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3 or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We make the complete list of such transformations up to birational symmetries. We also discuss correspondences of special solutions of Painlevé equations''. The method is based on the investigation of the Painlevé systems as Hamiltonian systems corresponding to the Painlevé equations. A folding transformation of such a system is an algebraic transformation which gives rise to a nontrivial quotient map of the space of initial conditions. Using these transformations, the authors give a further classification of the 6 Painlevé equations related to some Dynkin diagrams, lists of folding transformations and diagrams are given. See also the second author [Jap. J. Math., New Ser. 5, 1--79 (1979; Zbl 0426.58017)] and the third author [Commun. Math. Phys. 220, 165--229 (2001; Zbl 1010.34083].
scientific article; zbMATH DE number 2156242