Topological Obstacles for Complete Integrability

ThnSy the Kbrchhoif equations are nonintegrable in the general case. 

It will not be out of place to note that the integrable Clebsch case is determined just by the above condition  

The proof of Theorem 5.1.6 also rests upon the separatrix splitting phenomenon. To this end, one should represent the Kirchhoff equations as perturbations of integrable equations. Introduce a small parameter e replacing in the Kirchhoff equations e by ee. Then on the fixed four-dimensional level surface of two integrals 

1 Nonintegrability of the Equations of Motion of Natural Mechanical Systems with Two Degrees of Freedom on High-Genus Surfaces 

Consider the cotangent bundle T M to the manifold M. It is well known that the cotangent bundle T M of an arbitrary smooth manifold M can be 

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