to the constraint (\[eLNr1\]).

Applying the Bianchi identity to the both sides of (\[g.Ein.eqs\]), we obtain $$\label{BI}
\left(T_{\mu(mat)}^{\nu}+T_{\mu(obs)}^{\nu}+T_{\mu(ghost)}^{\nu}\right)_{;\nu}=0.$$ We can require that the conditions $T_{\mu ;\nu}^{\nu}=0$ would be satisfied separately by the energy-momentum tensor of matter fields and quasi-tensors of the observer and of ghosts. It will give us equations for the Lagrange multiplier $\pi_N$. Taking into account that $N^r=0$, $\pi_{N^r}=0$, the non-zero components of $T_{\mu(obs)}^{\nu}$ are $$\begin{aligned}
\label{obs00_1}
T^0_{0(obs)}&=&\frac1{4\pi}\dot\pi_N\frac1{VW^2};\\
\label{obs11_1}
T^1_{1(obs)}&=&-\frac1{4\pi}\dot\pi_N\frac1{NW^2}\frac{\partial f}{\partial V};\\
\label{obs22_1}
T^2_{2(obs)}=T^3_{3(obs)}&=&-\frac1{8\pi}\dot\pi_N\frac1{N V W}\frac{\partial f}{\partial W}.\end{aligned}$$ Equations $T_{\mu(obs);\nu}^{\nu}=0$ with $\mu=0, 1$ give $$\label{Tobs}
\frac d{dt}\left(\dot\pi_N N\right)=0,\qquad
\frac d{dr}\left(\dot\pi_N\frac{\partial f}{\partial V}V\right)=0.$$

The non-zero components of $T_{\mu(ghost)}^{\nu}$ are $$\begin{aligned}
\label{ghost00_1}
T^0_{0(ghost)}&=&\frac1{4\pi}\frac1{V W^2}\left[\dot{\bar\theta_0}(\theta^r)'
   -\dot{\bar\theta_0}\dot{\theta^0}\right];\\
\label{ghost11_1}
T^1_{1(ghost)}&=&\frac1{4\pi}\frac V{N W^2}\frac{\partial^2 f}{\partial V^2}\dot{\bar\theta_0}(\theta^r)';\\
\label{ghost22_1}
T^2_{2(ghost)}=T^3_{3(ghost)}&=&\frac1{8\pi}
   \left[\frac1{N W}\frac{\partial^2 f}{\partial V\partial W}\dot{\bar\theta_0}(\theta^r)'
    -\frac1{N V W}\frac{\partial f}{\partial W}\dot{\bar\theta_0}(\theta^r)'\right].\end{aligned}$$ As one can check, the equations $T_{\mu(ghost);\nu}^{\nu}=0$ are completely compatible with Lagrangian equations for ghosts.

### The Schwarzschild metric.

In this case the consideration of equations for ghost and the Lagrange multiplier $\pi_N$ does not lead to non-zero solutions for this variables, at least under requirements (\[tsol\]), (\[btsol\]), (\[piNr.sol\]). Indeed, Eqs. (\[eLt0r\_2\]) result in $\dot{\bar\theta_0}=0$, and all components of $T_{\mu(ghost)}^{\nu}$ are zero, thus the ghost sector gives no contribution. It follows from Eqs. (\[Tobs\]) that $$\label{pi.Schw}
\frac{\dot\pi_N}V={\rm 