&+&\frac1N\bar P_{\theta^0}P_{\bar\theta_0}
   +\bar P_{\theta^r}P_{\bar\theta_r}
   -\frac{N^r}N\bar P_{\theta^r}P_{\bar\theta_0}
   -\frac{N'}N\bar P_{\theta^0}\theta^r
   +N^r\bar P_{\theta^0}(\theta^0)'
   +N^r\bar P_{\theta^r}(\theta^r)'\nonumber\\
&-&(N^r)'\bar P_{\theta^r}\theta^r
   +\frac{N'N^r}N\bar P_{\theta^r}\theta^r
   +\frac{N^2}{V^2}\bar P_{\theta^r}(\theta^0)'
   +\frac{\partial f^r}{\partial V}V'\bar P_{\theta^r}\theta^r
   +\frac{\partial f^r}{\partial W}W'\bar P_{\theta^r}\theta^r\nonumber\\
&+&\frac{\partial f^r}{\partial V}V N^r\bar P_{\theta^r}(\theta^0)'
   +\frac{\partial f^r}{\partial V}V\bar P_{\theta^r}(\theta^r)'
   +\frac V N\frac{\partial f}{\partial V}\bar P_{\theta^0}(\theta^r)'
   +\frac{V N^r}N\frac{\partial f}{\partial V}\bar P_{\theta^0}(\theta^0)'\nonumber\\
&-&\frac{V N^r}N\frac{\partial f}{\partial V}\bar P_{\theta^r}(\theta^r)'
   -\frac{V(N^r)^2}N\frac{\partial f}{\partial V}\bar P_{\theta^r}(\theta^0)'
   +\frac{V'}N\frac{\partial f}{\partial V}\bar P_{\theta^0}\theta^r\nonumber\\
\label{Ham-EPS}
&-&\left.\frac{V'N^r}N\frac{\partial f}{\partial V}\bar P_{\theta^r}\theta^r
   +\frac{W'}N\frac{\partial f}{\partial W}\bar P_{\theta^0}\theta^r
   -\frac{W'N^r}N\frac{\partial f}{\partial W}\bar P_{\theta^r}\theta^r\right].\end{aligned}$$

The first line in (\[Ham-EPS\]) is the Hamiltonian for pure gravity that can be presented as a linear combination of Dirac secondary constraints since it is believed that a full derivative with respect to $r$ can be omitted in this expression: $$\label{Ham-D}
H_D=\int\limits_0^{\infty}dr\left[N\left(\frac1W P_V P_W-\frac V{2W^2}P_V^2
   -\frac{W W''}V-\frac{(W')^2}{2V}+\frac{V'W W'}{V^2}+\frac V2\right)
   +N^r\left(P_W W'-P'_V V\right)\right].$$ However, as it follows from (\[Ham-EPS\]), the Hamiltonian in extended phase space cannot be written down as a linear combination of constraints. Now we can write down the set of Hamiltonian equations in extended phase space presented explicitly in Appendix B.

It is important to emphasized that in this formulation of Hamiltonian dynamics the constraints as well as the gauge conditions have the status of Hamiltonian equations. Indeed, the Hamiltonian equations (\[DN\]), (\[DNr\]) coincide with 