  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.
