The standard Picart-Vessiot theory for differential
equations works well in the so-called Fuchs case , because Schlesinger
theorem connects algebra and topology . It does not work for such a
simple equation as dy = y.dx . We shall explain how we can solve this
difficulty by replacing a Galois group by a Galois groupoid , thus permitting
to develop a non-linear theory as Malgrange
and Umemura have done.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/3946