About rigorous methods of proving the non-integrability.

In this first part two methods of rigorous proving the non-integrability for dynamical systems will be presented. Both take their origin in works of Kovalevskaya and Lyapunov and their main idea lies in a deep connection between properties of solutions of the analysed system in the complex time plane and the existence of first integrals of this system.
More precisely, the first method due to Ziglin connects the existence of first integrals with properties of the monodromy group of variational equations. Its extension is the Morales-Ramis method that formulates the necessary integrability condition in the language of the differential Galois group.