BFV approach was originally proposed. However, we can think of it as a considerable mathematical indication that Lagrangian and Hamiltonian formalism appear to be non-equivalent for the full theory of gravity when one deals with spacetime manifolds of any topology, in particular, without asymptotic states which are implied in the $S$-matrix approach.

The central part in the BFV approach is given to the BRST charge constructed as a series in powers of Grassmannian variables with coefficients given by generalized structure functions of constraints algebra [@Hennaux]. Like the constraints, the BRST charge does not generates correct gauge transformations for all gravitational degrees of freedom including gauge ones. It is not surprising because the form of the BRST charge is determined by constraints algebra and, as was already mentioned, gauge transformations differ from those generated by the constraints.

The purpose of this paper is to present Hamiltonian dynamics in extended phase space which is free from the shortcomings mentioned above and can be thought of as a real alternative for Dirac generalized Hamiltonian dynamics, as well for the BFV formalism. The proposed approach has already been demonstrated for gravitational models with finite degrees of freedom ([@SSV1; @SSV2; @Shest1] and other papers). In 