Hamiltonian constrained formalism for the general field theories with unfree gauge symmetry

We consider constrained Hamiltonian formalism with any finite order of constraints for general field theory with unfree gauge symmetry. It is noticed that the redefinition of the Hamiltonian by adding constraint terms, as well as the linear combination of constraints of all generations, does not change the gauge identities, and therefore the unfree gauge symmetry transformations. The general formalism is exemplified by the model of linearized unimodular gravity, for which the triviality of a cubic vertex, proportional to the completion function of a free theory, is also demonstrated in the Hamiltonian formalism.