We prove that for any known Lie algebra g having none invariants for the coadjoint representation, the absence of invariants is equivalent to the existence of a left invariant exact symplectic structure on the corresponding Lie group G. We also show that a nontrivial generalized Casimir invariant constitutes an obstruction for the exactness of a symplectic form, and provide solid arguments to c...

It has been shown that the cases of the JWKB formulae in 1–dim QM quantizing the energy levels exactly are results of essentially one global symmetry of both potentials and their corresponding Stokes graphs. Namely, this is the invariance of the latter on translations in the complex plain of the space variable i.e. the potentials and the Stokes graphs have to be periodic. A proliferation of tur...