The first and the second Painlevé equations are explicitly Hamiltonian with time dependent Hamilton function. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems in ℂ⁴. We prove that the latter systems do not have any additional algebraic first integral. In the proof equations in variations with respect to a parameter are used.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc94-0-20,
author = {Henryk \.Zo\l \k adek},
title = {A simple proof of the non-integrability of the first and the second Painlev\'e equations},
journal = {Banach Center Publications},
volume = {95},
year = {2011},
pages = {295-302},
zbl = {1242.34156},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc94-0-20}
}
Henryk Żołądek. A simple proof of the non-integrability of the first and the second Painlevé equations. Banach Center Publications, Tome 95 (2011) pp. 295-302. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc94-0-20/