Proof of the Principal Classification Theorem

Any constant-energy surface of an integrable system can be represented as gluing the simple three-dimensional manifolds of three types. 

Let be an integrable constant-energy surface. Single out in all critical submanifolds of the integral /. Then have  

3) 9 0 saddle circles 5 for which the separatrix diagram is orient able  

5) r 0 maximal and minimal Klein bottles. Then Q admits the representation q = mU S ) + pU(T ) + rU K ) + qU S ) + sU S. ), where U L) denotes a connected regular tubular neighbourhood of the critical level surface containing the submanifold L (see above). In Lemma 2.1.9 all these manifolds are completely described  

The proof of Theorem 2.1.3 is obtained from Theorem 2.1.2 and from Lemma 2.1.10. 

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