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I'm searching for some references about groups of invariance of the Painlevé property other than the book of Robert Conte or more generally birational transformation of Riemann surfaces.

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I assume "the book of Robert Conte" refers to The Painlevé Handbook by Robert Conte and Micheline Musette (Springer, 2008); the "groups of invariance of the Painlevé property" are briefly described in Appendix A of that book.

The book by Harold T. Davis Introduction to Nonlinear Differential and Integral Equations (Dover, 1962) provides further background.

In this connection, the book by Einar Hille, Ordinary Differential Equations in the Complex Domain (Wiley, 1976; Dover reprint 1997) gives a different treatment and also provides historical references.

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