+N\dot\theta^0-N N^r(\theta^0)'-\frac{\partial f}{\partial V}V N^r(\theta^0)'
   -\frac{\partial f}{\partial V}V(\theta^r)';\\
\label{Pt0}
\bar P_{\theta^0}&=&\dot{\bar\theta_0}N+\dot{\bar\theta_r}N^r;\\
P_{\bar\theta_r}&=&N^r\dot\theta^0-\frac{N^2}{V^2}(\theta^0)'-(N^r)^2(\theta^0)'
   +\dot\theta^r-N^r(\theta^r)'+(N^r)'\theta^r\nonumber\\
\label{Pbtr}
&-&\frac{\partial f^r}{\partial V}V N^r(\theta^0)'
   -\frac{\partial f^r}{\partial V}V(\theta^r)'
   -\frac{\partial f^r}{\partial V}V'\theta^r
   -\frac{\partial f^r}{\partial W}W'\theta^r;\\
\label{Ptr}
\bar P_{\theta^r}&=&\dot{\bar\theta_r}.\end{aligned}$$

Introducing of the missing velocities by means of the differential form of gauge conditions (\[d.c.1\]), (\[d.c.2\]) enables us to construct a Hamiltonian in extended phase space not applying to the Dirac procedure, by the usual rule $$\label{Ham1}
H=\int\limits_0^{\infty}dr\left(P_N\dot N+P_{N^r}\dot N^r+P_V\dot V+P_W\dot W
   +\bar P_{\theta^0}\dot\theta^0+\dot{\bar\theta_0}P_{\bar\theta_0}
   +\bar P_{\theta^r}\dot\theta^r+\dot{\bar\theta_r}P_{\bar\theta_r}-L\right);$$ $$\begin{aligned}
H&=&\int\limits_0^{\infty}dr\left[\frac N W P_V P_W
   -\frac{N V}{2W^2}P_V^2+P_V V'N^r+P_V V(N^r)'+P_W W'N^r
   +\frac{N'W'W}V+\frac{N(W')^2}{2V}+\frac{N V}2\right.\nonumber\\
&+&P_N\frac{\partial f}{\partial V}V'N^r
   +P_N\frac{\partial f}{\partial W}W'N^r
   +P_N\frac{\partial f}{\partial V}V(N^r)'
   +P_{N^r}\frac{\partial f^r}{\partial V}V'N^r
   +P_{N^r}\frac{\partial f^r}{\partial W}W'N^r\nonumber\\
&+&P_{N^r}\frac{\partial f^r}{\partial V}V(N^r)'
   +\frac N W P_V P_N\frac{\partial f}{\partial W}
   +\frac N W P_W P_N\frac{\partial f}{\partial V}
   -\frac{N V}{W^2}P_V P_N\frac{\partial f}{\partial V}\nonumber\\
&+&\frac N W P_V P_{N^r}\frac{\partial f^r}{\partial W}
   +\frac N W P_W P_{N^r}\frac{\partial f^r}{\partial V}
   -\frac{N V}{W^2}P_V P_{N^r}\frac{\partial f^r}{\partial V}
   -\frac{N V}{2W^2} P_N^2\left(\frac{\partial f}{\partial V}\right)^2\nonumber\\
&-&\frac{N V}{2W^2}P_{N^r}^2\left(\frac{\partial f^r}{\partial V}\right)^2
   +\frac N W P_N^2\frac{\partial f}{\partial V}
    \frac{\partial f}{\partial W}
   +\frac N W P_{N^r}^2\frac{\partial f^r}{\partial V}
    \frac{\partial f^r}{\partial W}\nonumber\\
&+&\frac N W P_N P_{N^r}\frac{\partial f}{\partial V}
    \frac{\partial f^r}{\partial W}
   +\frac N W P_N P_{N^r}\frac{\partial f}{\partial W}
    \frac{\partial f^r}{\partial V}
   -\frac{N V}{W^2}P_N P_{N^r}\frac{\partial f}{\partial V}
    \frac{\partial f^r}{\partial V}\nonumber\\
