Projective Differential Geometrical Structure of the Painlevé Equations

Symmetry group classification of ordinary differential equations: Survey of some results

“…This approach was also employed for obtaining descriptions of the equivalence classes to the first and second Painlevé transcendents [58]. In this context, it is worth mentioning the works [59,60] in which equivalence to Painlev'e equations are also observed. Moreover, Leach and co-workers have also employed the Painlevé analysis (see, e.g.…”

On a surface isolated by Gambier

“…Consequently, all solutions of (10) are transcendental. 1 Algebraic solutions of (13) [8] are in fact rational solutions for another representative of P III in its equivalence class under (u, x) → (g(x)u, f (x)), with f (x) = √ x, g(x) = 1. All algebraic solutions of Pn equations, n = 2, 3, 4, 5, are similarly rational.…”

Open Problems for Painlevé Equations

“…where F is rational in w and dw/dz and locally analytic in z and solved the equations in terms of the first, second and fourth Painlevé transcendents, elliptic functions, or quadratures. For various results on classifying classes of second-order ordinary differential equations, including Painlevé equations, see Babich and Bordag [12], Bagderina [13,15,16,17,18], Bagderina and Tarkhanov [19], Berth and Czichowski [22], Hietarinta and Dryuma [78], Kamran, Lamb and Shadwick [88], Kartak [90,91,92,93], Kossovskiy and Zaitsev [97], Milson and Valiquette [106], Valiquette [139] and Yumaguzhin [145]. Most of these studies are concerned with the invariance of second-order ordinary differential equations of the form…”