  From the viewpoint of the SU(2) coherent states (CS) and their path integrals
(PI) labeled by a full set of Euler angles $(\phi, \theta, \psi)$ which we
developed in the previous paper, we study the relations between gauge
symmetries of Lagrangians and allowed quantum states; we investigate
permissible types of fiducial vectors (FV) in the full quantum dynamics in
terms of SU(2) coherent states for typical Lagrangians. We propose a general
framework for a Lagrangian having a certain gauge symmetry with respect to one
of the Euler angles $\psi$. We find that for the case fiducial vectors are so
restricted that they belong to the eigenstates of ${\hat S}_3$ or to the orbits
of them under the action of the SU(2); and the strength of a fictitious
monopole, which appears in the Lagrangian, is a multiple of $\frac12$. In this
case Dirac strings are permitted. Our formulations and results deepen those of
the preceding work by Stone that has piloted us; we illustrate the relation
between the two methods. The reasoning here does not work for a Lagrangian
without the gauge symmetry. This suggests a new possibility about monopole
charge quantization. Besides analogies to field theory and entanglements in
quantum information (QI) are briefly mentioned.
