Quantum Hamiltonian Systems on K-Orbits: Semiclassical Spectrum of the Asymmetric Top

We consider equations on Lie groups and classical and quantum Hamiltonian systems on coadjoint representation orbits. We show that the transition to canonical coordinates on orbits of the coadjoint representation allows constructing semiclassical solutions and the corresponding spectra of quantum equations such that all the symmetries of the original problem are preserved. Our method is used to find the semiclassical spectrum of the asymmetric quantum top.