Geodesic
Bäcklund transformations from painlevé analysis
Glasgow mathematical journal, Tome 43 (2001) no. A, pp. 9-21

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Since its elaboration in 1983 by Weiss, Tabor and Carnevale, the method to explicitly build the Bäcklund transformation of a partial differential equation (PDE) from singularity analysis only has been improved in several complementary directions, and at the present time it succeeds for practically all PDEs in 1+1-dimensions. The current state of the art is presented, and the emphasis is put on understanding the method. There are two important stages: first, the definition (identified with a Darboux transformation) of a resummation variable to make the Laurent series a finite one as requested by the definition of the word integrability; second, the link (identified with a linearizing formula to be taken from the classification of Painlevé and Gambier) between this resummation variable and the Lax pair to be found. 
Conte, Robert; Musette, Micheline. Bäcklund transformations from painlevé analysis. Glasgow mathematical journal, Tome 43 (2001) no. A, pp. 9-21. doi: 10.1017/S0017089501000027
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