Umemura, Hiroshi On the transformation group of the second Painlevé equation. (English) Zbl 0972.34073 Nagoya Math. J. 157, 15-46 (2000). For the second Painlevé equation \[ y''=2y^3+ty +\alpha, \quad \alpha\in\mathbb C, \] the Bäcklund transformation group is isomorphic to the extended affine Weyl group of type \(\widetilde{A}_1,\) namely the group \(G\) generated by the translations \(t_+(\alpha)=\alpha+1,\) \(t_-(\alpha)=\alpha-1\) and the reflection \(i(\alpha)=-\alpha.\) For the affine spaces \(\mathbb A^4\) and \(\mathbb A^2\) with coordinate systems \((y,y',t,\alpha)\) and \((t,\alpha),\) respectively, consider the vector field \[ \delta(\alpha)=\frac{\partial}{\partial t}+y' \frac{\partial}{\partial y}+(2y^3+ty+\alpha) \frac{\partial}{\partial y'} \] on \(\mathbb A^4\) and a natural fibration \(\pi : \mathbb A^4 \to \mathbb A^2\) by \((y,y',t,\alpha)\mapsto (t,\alpha).\) Here, the author constructs a projective model of the fibration \(\pi : \mathbb A^4 \to \mathbb A^2\) on which the group \(G\) operates regularly. Reviewer: Shun Shimomura (Yokohama) Cited in 2 Documents MSC: 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies Keywords:Bäcklund transformation; Weyl group; vector field; projective model; fibration PDFBibTeX XMLCite \textit{H. Umemura}, Nagoya Math. J. 157, 15--46 (2000; Zbl 0972.34073) Full Text: DOI Digital Library of Mathematical Functions: §32.7(viii) Affine Weyl Groups ‣ §32.7 Bäcklund Transformations ‣ Properties ‣ Chapter 32 Painlevé Transcendents References: [1] Jap. J. Math 5 pp 1– (1979) [2] Funkcial. Ekvac 28 pp 1– (1985) [3] J. Math. Soc. Japan [4] DOI: 10.1007/BF01458459 · Zbl 0589.58008 · doi:10.1007/BF01458459 [6] Funkcial. Ekvac 40 pp 271– (1997) [7] Nagoya Math. J 148 pp 151– (1997) · Zbl 0934.33029 · doi:10.1017/S0027763000006486 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.