Andrzej J. Maciejewski 氏
(University of Zielona Góra, Poland)
I will present applications of several methods of proving non-integrability of differential equations which grow from the idea of Kovalevskaya: integrability of differential equations implies that their solutions are single valued. From this idea arose the Painlev\'e analysis and Ziglin theory which base on analytical tools. The same idea gives rise also purely algebraic Lagutynsky-Levelt method which together with Darboux method is frequently used for proving non-integrability. Finally, the Ziglin theory has found its algebraic version in the Marales-Ramis theory. I give several examples which show weak and strong aspects of analytic and algebraic approaches for proving non-integrability.