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The Morales-Ramis theory provides an effective and powerful non-integrabilitycriterion for complex analytical Hamiltonian systems via the differentialGaloisian obstruction. In this paper we give a new Morales-Ramis type theoremon the meromorphic Jacobi non-integrability of general analytic dynamicalsystems. The key point is to show the existence of Jacobian multiplier of anonlinear system implies the existence of common Jacobian multiplier of Liealgebra associated with the identity component. In addition, we apply ourresults to the polynomial integrability of Karabut systems for stationarygravity waves in finite depth.

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