Ordinary differential equations all share the same common root-real physical problems. But, although the physical motivation remains the most important one, the way the subject develops does depend highly on the methods available. In the exposition I would like to show some connections between two methods of checking the ODE for integrability (whatever it should mean), with distant motivations and techniques. These are the so-called Painlevé tests and the methods originating in Ziglin's theory and developed widely by Morales and Ramis.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc58-0-5,
author = {Pawe\l\ Goldstein},
title = {Kovalevska vs. Kovacic-two different notions of integrability and their connections},
journal = {Banach Center Publications},
volume = {58},
year = {2002},
pages = {63-73},
zbl = {1043.34099},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc58-0-5}
}
Paweł Goldstein. Kovalevska vs. Kovacic-two different notions of integrability and their connections. Banach Center Publications, Tome 58 (2002) pp. 63-73. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc58-0-5/