Geodesic
Truncation-type methods and Bäcklund transformations for ordinary differential equations: the third and fifth Painlevé equations
Glasgow mathematical journal, Tome 43 (2001) no. A, pp. 23-32

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In a recent paper we presented a truncation-type method of deriving Bäcklund transformations for ordinary differential equations. This method is based on a consideration of truncation as a mapping that preserves the locations of a natural subset of the movable poles that the equation possesses. Here we apply this approach to the third and fifth Painlevé equations. For the third Painlevé equation we are able to obtain all fundamental Bäcklund transformations for the case where the parameters satisfy \gamma \delta \neq 0. For the fifth Painlevé equation our approach yields what appears to be all known Bäcklund transformations. 
Gordoa, P. R.; Joshi, N.; Pickering, A. Truncation-type methods and Bäcklund transformations for ordinary differential equations: the third and fifth Painlevé equations. Glasgow mathematical journal, Tome 43 (2001) no. A, pp. 23-32. doi: 10.1017/S0017089501000039
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     title = {Truncation-type methods and {B\"acklund} transformations for ordinary differential equations: the third and fifth {Painlev\'e} equations},
     journal = {Glasgow mathematical journal},
     pages = {23--32},
     year = {2001},
     volume = {43},
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     doi = {10.1017/S0017089501000039},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089501000039/}
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