  We provide geometric quantization of a completely integrable Hamiltonian
system in the action-angle variables around an invariant torus with respect to
the angle polarization. The carrier space of this quantization is the
pre-Hilbert space of smooth complex functions on the torus. A Hamiltonian of a
completely integrable system in this carrier space has a countable spectrum. If
it is degenerate, its eigenvalues are countably degenerate. We study
nonadiabatic perturbations of this Hamiltonian by a term depending on classical
time-dependent parameters. It is originated by a connection on the parameter
space, and is linear in the temporal derivatives of parameters. One can choose
it commuting with a degenerate Hamiltonian of a completely integrable system.
Then the corresponding evolution operator acts in the eigenspaces of this
Hamiltonian, and is an operator of parallel displacement along a curve in the
parameter space.
