  Zero modes of first class secondary constraints in the two-dimensional
electrodynamics and the four-dimensional SU(2) Yang-Mills theory are considered
by the method of reduced phase space quantization in the context of the problem
of a stable vacuum. We compare the description of these modes in the Dirac
extended method and reveal their connection with the topological structure of
the gauge symmetry group. Within the framework of the "reduced" quantization we
construct a new global realization of the homotopy group representation in the
Yang-Mills theory, where the role of the stable vacuum with a finite action
plays the Prasad-Sommerfield solution.
