not invariant under gauge transformation and, therefore under BRST transformations (\[delN1\]) – (\[delW1\]). Then, we should return to the gravitational action (\[grav-act1\]). Now we deal with the Lagrangian which involves second derivatives of metric components and ghosts. The BRST charge is constructed in accordance with the Noether theorem generalized for theories with high derivatives: $$\label{Noet.BRST}
\Omega=\int d^3x\left[\frac{\partial L}{\partial(\partial_0\phi^a)}\delta\phi^a
   +\frac{\partial L}{\partial(\partial _0\partial_{\mu}\phi^a)}\delta(\partial_{\mu}\phi^a)
   -\partial_{\mu}\left(\frac{\partial L}{\partial(\partial_0\partial_{\mu}\phi^a)}\right)\delta\phi^a
   +\partial _0\left(L x^0\right)\right].$$ $\phi^a$ stands for all variables $N$, $N^r$, $V$, $W$ and ghosts. After some tedious calculations we come to the following expression for the BRST charge in the spherically symmetric model: $$\begin{aligned}
\Omega&=&\int\!dr\left[-{\cal H}\theta^0-P_V V'\theta^r
   -P_N\frac{\partial f}{\partial V}V'\theta^r
   -P_{N^r}\frac{\partial f^r}{\partial V}V'\theta^r\right.\nonumber\\
&-&P_W W'\theta^r-P_N\frac{\partial f}{\partial W}W'\theta^r
   -P_{N^r}\frac{\partial f^r}{\partial W}W'\theta^r\nonumber\\
&-&P_V V N^r(\theta^0)'-P_N\frac{\partial f}{\partial V} V N^r(\theta^0)'
   -P_{N^r}\frac{\partial f^r}{\partial V} V N^r(\theta^0)'\nonumber\\
&-&P_V V(\theta^r)'-P_N\frac{\partial f}{\partial V} V(\theta^r)'
   -P_{N^r}\frac{\partial f^r}{\partial V} V(\theta^r)'\nonumber\\
\label{BRST1}
&-&\left.\bar P_{\theta^0}(\theta^0)'\theta^r
   -\bar P_{\theta^r}(\theta^r)'\theta^r
   -P_N P_{\bar\theta_0}-P_{N^r}P_{\bar\theta_r}
   -\frac{N W W'(\theta^0)'}V\right],\end{aligned}$$ $\cal H$ is a Hamiltonian density in (\[Ham-EPS\]). It can be directly checked that the charge (\[BRST1\]) generates transformations (\[delN1\]) – (\[delbtr\]). Let us emphasized that the Hamiltonian equations in extended phase space, in particular, constraints and gauge conditions which have the status of Hamiltonian equations, is used to get correct results, for instance, $$\begin{aligned}
\delta 