Geodesic
Quadratic transformations for the third and fifth Painlev\'e equations
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 18, Tome 317 (2004), pp. 105-121

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Quadratic transformations for the third and fifth Painlevé equations are constructed via the method of $RS$-transformations. This method can be viewed as a prolongation of the quadratic transformations for the Painlevé equations to the associated linear ODEs, whose isomonodromy deformations are governed by the corresponding Painlevé equations. 
@article{ZNSL_2004_317_a6,
     author = {A. V. Kitaev},
     title = {Quadratic transformations for the third and fifth {Painlev\'e} equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {105--121},
     publisher = {mathdoc},
     volume = {317},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a6/}
}
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A. V. Kitaev. Quadratic transformations for the third and fifth Painlev\'e equations. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 18, Tome 317 (2004), pp. 105-121. http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a6/