Geodesic
Necessary conditions of point equivalence of second-order ODEs to the sixth Painlev\'e equation
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 25, Tome 473 (2018), pp. 17-33

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Equivalence problem for a projective type scalar second-order ordinary differential equations is considered with respect to invertible point changes of variables. Invariants of the equivalence transformation group of this family of equations are used to find the necessary conditions of the equivalence to the sixth Painlevé equation. 
@article{ZNSL_2018_473_a1,
     author = {Yu. Yu. Bagderina},
     title = {Necessary conditions of point equivalence of second-order {ODEs} to the sixth {Painlev\'e} equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {17--33},
     publisher = {mathdoc},
     volume = {473},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_473_a1/}
}
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Yu. Yu. Bagderina. Necessary conditions of point equivalence of second-order ODEs to the sixth Painlev\'e equation. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 25, Tome 473 (2018), pp. 17-33. http://geodesic.mathdoc.fr/item/ZNSL_2018_473_a1/