Dirac approach only $g_{ij}$ components with their conjugate momenta are included into phase space, the transformations for this variables are generated by constraints. The constraints, which do not depend on gauge variables, play a crucial role in quantization procedure, while gauge variables are dropped out of further consideration.

It is well-known that the Wheeler – DeWitt quantum geometrodynamics [@DeWitt], constructed by a direct application of the Dirac quantization scheme to gravity, has faced quite a number of problems (see, for example, [@Isham; @SS1]). At the same time, modern approaches to quantization of gravity, such as Loop Quantum Gravity, keep considering the Wheeler – DeWitt equation as a main constituent of the theory [@Rovelli].

It is worth mentioning that the algebra of constraints in original phase space not including gauge degrees of freedom does depend on parametrization of gravitational variables. As was argued in our previous paper [@Shest1], it creates a serious obstacle to find an algorithm to construct a generator that would give correct transformations for all gravitational variables in the limits of the Dirac approach. Moreover, any transformations of gravitational variables, which touch upon gauge degrees of freedom, are not canonical from the viewpoint of the Dirac formalism. Even 