Differential Galoisian approach to Jacobi integrability of general analytic dynamical systems and its application

The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytical Hamiltonian systems via the differential Galoisian obstruction. In this paper we give a new Morales-Ramis type theorem on the meromorphic Jacobi non-integrability of general analytic dynamical systems. The key point is to show the existence of Jacobian multiplier of a nonlinear system implies the existence of common Jacobian multiplier of Lie algebra associated with the identity component. In addition, we apply our results to the polynomial integrability of Karabut systems for stationary gravity waves in finite depth.