fixing the gauge condition which helps to separate out the required solution, meanwhile usually a gauge is imposed on a final stage of the procedure. So we get the Schwarzschild metric: $$\label{Schw.met}
ds^2=-\left(1-\frac{2GM}r\right)dt^2+\frac{dr^2}{1-\displaystyle\frac{2GM}r}
   +r^2\left(d\theta^2+\sin^2\theta d\varphi^2\right).$$

### The Lemaitre –- Tolman – Bondi metric.

The Lemaitre –- Tolman – Bondi solution is the solution for dust matter in synchronous and comoving reference frame (see, for example, [@Enqvist] and references therein). Actually one should add energy-momentum tensor of dust into the equations. In this case the only non-zero component of the matter energy-momentum tensor is $T_{0(mat)}^0=\varepsilon$, where $\varepsilon$ is the energy density of dust. After imposing the gauge condition $$\label{g.c.1.3}
N=1$$ Eq. (\[eLN1\]) can be rewritten as $$\label{eLN3}
\frac{2\dot V\dot W}{V W}+\frac{\dot W^2}{W^2}+\frac1{W^2}
   -\frac{(W')^2}{V^2W^2}-\frac{2W''}{V^2W}+\frac{2V'W'}{V^3W}
   =8\pi\varepsilon.$$ Eq. (\[eLNr1\]) gives $$\label{eLNr3}
0=W'\dot V-V\dot W'.$$ Its solution yields a relation between $V(t,r)$ and $W(t,r)$, $$\label{LTB.sol}
V(t,r)=C_1(r)W'(t,r),$$ where $C_1(r)$ is an arbitrary function. The solution (\[LTB.sol\]) can be obtained from Hamiltonian equations. From (\[DW1\]), (\[DV1\]) one can express $$\label{LTB.mom}
P_V=W\dot W,\qquad
P_W=W\dot V+V\dot W.$$ Substitution (\[LTB.mom\]) into (\[DPNr1\]) gives (\[LTB.sol\]). Finally, the Lemaitre –- Tolman – Bondi metric can be presented as $$\label{LTB.met}
ds^2=-dt^2+\left(C_1(r)W'(t,r)\right)^2 dr^2
   +W^2(t,r)\left(d\theta^2+\sin^2\theta d\varphi^2\right).$$

Eq. (\[eLV1\]) after substitution (\[LTB.sol\]) leads to $$\label{eLV3}
\dot 