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The unfree gauge symmetry implies that gauge variation of the actionfunctional vanishes provided for the gauge parameters are restricted by thedifferential equations. The unfree gauge symmetry is shown to lead to theglobal conserved quantities whose on shell values are defined by theasymptotics of the fields or data on the lower dimension surface, or even atthe point of the space-time, rather than Cauchy hyper-surface. The most knownexample of such quantity is the cosmological constant of unimodular gravity.More examples are provided in the article for the higher spin gravity analoguesof the cosmological constant. Any action enjoying the unfree gauge symmetry isdemonstrated to admit the alternative form of gauge symmetry with the higherorder derivatives of unrestricted gauge parameters. The higher order gaugesymmetry is reducible in general, even if the unfree symmetry is not. Therelationship is detailed between these two forms of gauge symmetry in theconstrained Hamiltonian formalism. The local map is shown to exist from theunfree gauge algebra to the reducible higher order one, while the inverse mapis non-local, in general. The Hamiltonian BFV-BRST formalism is studied forboth forms of the gauge symmetry. These two Hamiltonian formalisms are shownconnected by canonical transformation involving the ghosts. The generatingfunction is local for the transformation, though the transformation as such isnot local, in general. Hence, these two local BRST complexes are notquasi-isomorphic in the sense that their local BRST-cohomology groups can bedifferent. This difference in particular concerns the global conservedquantities. From the standpoint of the BRST complex for unfree gauge symmetry,these quantities are BRST-exact, while for the alternative complex, thesequantities are the non-trivial co-cycles.

Reducible gauge symmetry versus unfree gauge symmetry in Hamiltonian formalism