Orientable Bott integral

Definition 2.1.3 We say that a Bott integral / on a surface Q is orientable if all of its critical submanifolds are orientable. If at least one of its critical submanifolds is nonorientable, we say that the integral / is nonorientable. 

FVom this it follows that if / is a nonorientable integral on Q, then (Q) 0, and the group xi(Q) contains a subgroup of index two. If, for instance, Q is homeomorphic to the sphere (a particular case in mechanics), then any Bott integral / on 5 is always orientable. 

Corollary 2.1.4. Not nearly each three-dimensional smooth compact closed orientable manifold may play the role of a constant-energy surface of a Hamiltonian system integrated by means of a smooth Bott integral. 

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