form $$\label{g.c.2.1}
N^r=0.$$

Taking into account (\[g.c.2.1\]) we come to the following set of equations: $$\begin{aligned}
\label{eLN1}
0&=&\frac{\dot V W\dot W}{N^2}+\frac{V\dot W^2}{2N^2}
   +\frac V2-\frac{(W')^2}{2V}-\frac{W W''}V+\frac{V'W'W}{V^2};\\
\label{eLNr1}
0&=&\frac{W W'\dot V}N-\frac{W V\dot W'}N+\frac{N'WV\dot W }{N^2};\\
\label{eLV1}
0&=&\frac{\dot W^2}{2N}+\frac{W\ddot W}N-\frac{W\dot W\dot N}{N^2}+\frac N2
   -\frac{N(W')^2}{2V^2}-\frac{N'W'W}{V^2};\\
\label{eLW1}
0&=&\frac{W\ddot V}N+\frac{\dot V\dot W}N+\frac{V\ddot W}N-\frac{\dot N\dot V W}{N^2}
   -\frac{\dot N\dot W V}{N^2}-\frac{W N''}V+\frac{W N'V'}{V^2}
   -\frac{W'N'}V+\frac{W'N V'}{V^2}-\frac{W''N}V;\\
\label{g.c.1.1}
0&=&N-f(V,W).\end{aligned}$$

Solutions to Eqs. (\[eLN1\]) – (\[eLW1\]) can be considered as solutions of gauge invariant Einstein equations for the spherically symmetric model, though any solution requires fixing of gauge conditions to be presented in its final form. Eqs. (\[eLN1\]), (\[eLV1\]), (\[eLW1\]) can be obtained from a Lagrangian for the model with the metric (\[mod.int\]) where $N^r(t, r)=0$. Let us note that Eq. (\[eLNr1\]) which is equivalent to the $\left(0\atop 1\right)$ Einstein equation could not be obtained if one just substitutes (\[g.c.2.1\]) to the gravitational part of the action. The second gauge condition (\[g.c.1.1\]) can be imposed on different stages of solving the equations, but in what following we shall start from fixing this condition.

The Hamiltonian set of equations will be also rather simplified: $$\begin{aligned}
\label{DN1}
\dot N&=&\frac N W P_V\frac{\partial f}{\partial W}
   +\frac N W P_W\frac{\partial f}{\partial V}
   -\frac{N V}{W^2}P_V\frac{\partial 