  The unfree gauge symmetry implies that gauge variation of the action
functional vanishes provided for the gauge parameters are restricted by the
differential equations. The unfree gauge symmetry is shown to lead to the
global conserved quantities whose on shell values are defined by the
asymptotics of the fields or data on the lower dimension surface, or even at
the point of the space-time, rather than Cauchy hyper-surface. The most known
example of such quantity is the cosmological constant of unimodular gravity.
More examples are provided in the article for the higher spin gravity analogues
of the cosmological constant. Any action enjoying the unfree gauge symmetry is
demonstrated to admit the alternative form of gauge symmetry with the higher
order derivatives of unrestricted gauge parameters. The higher order gauge
symmetry is reducible in general, even if the unfree symmetry is not. The
relationship is detailed between these two forms of gauge symmetry in the
constrained Hamiltonian formalism. The local map is shown to exist from the
unfree gauge algebra to the reducible higher order one, while the inverse map
is non-local, in general. The Hamiltonian BFV-BRST formalism is studied for
both forms of the gauge symmetry. These two Hamiltonian formalisms are shown
connected by canonical transformation involving the ghosts. The generating
function is local for the transformation, though the transformation as such is
not local, in general. Hence, these two local BRST complexes are not
quasi-isomorphic in the sense that their local BRST-cohomology groups can be
different. This difference in particular concerns the global conserved
quantities. From the standpoint of the BRST complex for unfree gauge symmetry,
these quantities are BRST-exact, while for the alternative complex, these
quantities are the non-trivial co-cycles.
