charge in accordance with the Noether theorem (see Section 4).

The gauge-fixing part of the action is $$\label{gfix-act1}
S_{(gauge)}
 =\int dt\int\limits_0^{\infty}dr\left[\lambda_0\left(\dot N-\frac{\partial f}{\partial V}\dot V
   -\frac{\partial f}{\partial W}\dot W\right)
  +\lambda_r\left(\dot N^r-\frac{\partial f^r}{\partial V}\dot V
   -\frac{\partial f^r}{\partial W}\dot W\right)\right].$$

Taking into account gauge transformations for gravitational variables $$\begin{aligned}
\label{delN}
\delta N&=&-\dot N\eta^0-N'\eta^r-N\dot\eta^0+N N^r(\eta^0)';\\
\label{delNr}
\delta N^r&=&-\dot N^r\eta^0-(N^r)'\eta^r-N^r\dot\eta^0-\dot\eta^r
   +N^r(\eta^r)'+\frac{N^2}{V^2}(\eta^0)'+(N^r)^2(\eta^0)';\\
\label{delV}
\delta V&=&-\dot V\eta^0-V'\eta^r-V(\eta^r)'-V N^r(\eta^0)';\\
\label{delW}
\delta W&=&-\dot W\eta^0-W'\eta^r,\end{aligned}$$ that follow from (\[g\_transf\]), we get the Faddeev – Popov ghost action in the form $$\begin{aligned}
\label{ghost-act1}
S_{(ghost)}&=&\int dt\int\limits_0^{\infty}dr
   \left[\bar\theta_0\frac{d}{dt}\left(-\dot N\theta^0-N'\theta^r-N\dot\theta^0
    +N N^r(\theta^0)'\right.\right.\nonumber\\
&-&\left.\frac{\partial f}{\partial V}\left[-\dot V\theta^0-V'\theta^r-V(\theta^r)'-V N^r(\theta^0)'\right]
    -\frac{\partial f}{\partial W}\left[-\dot W\theta^0-W'\theta^r\right]\right)\nonumber\\
&+&\bar\theta_r\frac{d}{dt}\left(-\dot N^r\theta^0-(N^r)'\theta^r-N^r\dot\theta^0-\dot\theta^r
    +N^r(\theta^r)'+\frac{N^2}{V^2}(\theta^0)'+(N^r)^2(\theta^0)'\right.\nonumber\\
&-&\left.\left.\frac{\partial f^r}{\partial V}\left[-\dot V\theta^0-V'\theta^r-V(\theta^r)'-V N^r(\theta^0)'\right]
    -\frac{\partial f^r}{\partial W}\left[-\dot W\theta^0-W'\theta^r\right]\right)\right]\end{aligned}$$ $\bar\theta_0$, $\theta^0$, $\bar\theta_r$, $\theta^r$ are ghost variables. After redefinition $$\label{lambda}
\pi_N=\lambda_0+\dot{\bar\theta_0}\theta^0;\quad
\pi_{N^r}=\lambda_r+\dot{\bar\theta_r}\theta^0$$ we can write the effective Lagrangian in the form without second derivatives: $$\begin{aligned}
\label{full-act2}
S_{(eff)}&=&\int dt\int\limits_0^{\infty}dr\left(\frac{\dot V\dot W W}N
  +\frac{V\dot W^2}{2N}
  -\frac{N'W'W}V
  -\frac{N(W')^2}{2V}
  -\frac{NV}2\right.\nonumber\\
 &-&\frac{W'\dot WVN^r}N
  -\frac{W W'\dot V N^r}N
  -\frac{W\dot W V'N^r}N
  -\frac{W\dot W V(N^r)'}N\nonumber\\
 &+&\frac{W W'V'(N^r)^2}N
  + \frac{W W'V N^r(N^r)'}N
  +\frac{(W')^2 V(N^r)^2}{2N}\nonumber\\
 &+&\pi_N\left(\dot N-\frac{\partial f}{\partial V}\dot V-\frac{\partial f}{\partial W}\dot W\right)
  +\pi_{N^r}\left(\dot N^r-\frac{\partial f^r}{\partial V}\dot V
   -\frac{\partial f^r}{\partial W}\dot W\right)\nonumber\\
 &+&\dot{\bar\theta_0}\theta^r\left(N'-\frac{\partial f}{\partial V}V'
   -\frac{\partial f}{\partial W}W'\right)\nonumber\\
 &+&\dot{\bar\theta_0}\left(N\dot\theta^0-NN^r(\theta^0)'-\frac{\partial f}{\partial V}VN^r(\theta^0)'
   -\frac{\partial f}{\partial V}V(\theta^r)'\right)\nonumber\\
 &+&\dot{\bar\theta_r}\left[N^r\dot\theta^0-\left(\frac{N^2}{V^2}+(N^r)^2\right)(\theta^0)'+\dot\theta^r
   -N^r(\theta^r)'+(N^r)'\theta^r\right.\nonumber\\
 &-&\left.\left.\frac{\partial f^r}{\partial V}\left(VN^r(\theta^0)'+V(\theta^r)'+V'\theta^r\right)
   