ABSTRACT
The Ziglin theory relates the integrability of the original system to the appropriately defined integrability of variational equations around a particular solution. It was applied for proving the non-integrability of many problems connected with rigid body dynamics, orbital dynamics and cosmological models. As in the Ziglin theory, the authors consider a complex Hamiltonian system for which they know a particular solution. In the Ziglin theory they ask when the system with n degrees of freedom possesses n, not necessarily commuting, first integrals. In the Morales-Ramis theory they ask whether the system is completely integrable. Moreover, in the Morales-Ramis theory they ask how the assumed integrability manifests itself by properties of the differential Galois group. These differences are important because on the one hand the imposed integrability is more restrictive than in the Ziglin theory, and on the other hand the differential Galois group is bigger than the monodromy group, thus, in general, it must be easier to prove the non-integrability.
