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Bäcklund transformations and the Painlevé property. (English) Zbl 0738.58052
Lie theory, differential equations and representation theory, Proc. Annu. Semin. Can. Math. Soc., Montréal/Can. 1989, 405-428 (1990).
[For the entire collection see Zbl 0728.00010.]
A partial differential equation is said to have the Painlevé property if the solutions of the PDE are single valued about the movable singularity manifolds. To verify this property, the author and others have introduced the “singular manifold method” [see the author, J. Math. Phys. 24, 1405-1413 (1983; Zbl 0531.35069)], which is also outlined in the present paper. The power of this method is demonstrated by a detailed analysis of the Korteweg-de Vries equation. For systems which do not have the Painlevé property, the author introduces a concept of conditional integrability, based on constrained expansions. He formulates the following
Conjecture: The constraint equations are completely integrable.
In the referee’s opinion, this is an exciting paper, full of ideas, important examples and applications.

MSC:
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35Q53 KdV equations (Korteweg-de Vries equations)