  Using extended Schwinger's quantization approach quantum mechanics on a
Riemannian manifold $M$ with a given action of an intransitive group of
isometries is developed. It was shown that quantum mechanics can be determined
unequivocally only on submanifolds of $M$ where $G$ acts simply transitively
(orbits of $G$-action). The remaining part of degrees of freedom can be
described unequivocally after introducing some additional assumptions. Being
logically unmotivated, these assumptions are similar to canonical quantization
postulates. Besides this ambiguity that has a geometrical nature there is
undetermined gauge field of $\hbar$ (or higher) order, vanishing in the
classical limit $\hbar\to 0$.
