this work we apply our approach to generalized spherically symmetric gravitational model which imitates the full gravitational theory much better, so that one can see the way how one can get appropriate results in the case of the full theory. In Section 2 the Lagrangian and Hamiltonian dynamics in extended phase space are derived from the effective action for the model, and the structure of the Hamiltonian function and Hamiltonian equations is analyzed. In Section 3 we shall consider some particular cases that the generalized model embraces, and show that the solutions to the obtained equations comprise solutions to the gauge invariant Einstein equations with corresponding symmetry. We also discuss a possible role of gauge-noninvariant terms. In Section 4 we shall make use of BRST invariance of effective action and construct the BRST charge according to the Noether theorem which generates correct transformations for all gravitational degrees of freedom. The proposed method will be shown to be self-consistent, and the equivalence of the Lagrangian and Hamiltonian formulations can be proved by direct calculations.

The model, its Lagrangian and Hamiltonian dynamics
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In this paper we shall follow to the ADM parametrization for space-time metric: $$\label{gen.int}
ds^2=\left(-N^2+N_i N^i\right)dt^2+2N_i dt dx^i+g_{ij}dx^i dx^j.$$ 