in the synchronous reference frame reads as $$\label{metricaBI}
ds^{2}=-dt^{2}+R_1(t)^2\,dx^{2}+R_2(t)^2\,dy^{2}+R_3(t)^2\,dz^{2}$$ and hence the metric determinant is given by the following relation $$\sqrt{-g}=R_1 R_2 R_3\equiv R^{3}.$$ Above, $x, y, z$ denote Euclidean coordinates and $R_1(t)$, $R_2(t)$ and $R_3(t)$ are dubbed cosmic scale factors. As well-known (see [@Montani],[@Landau1] and [@Gravitation]) such a model describes in vacuum an intrinsic anisotropic Universe, but in the presence of matter it can also admit the isotropic limit [@KirillovMo]. Let us now introduce the following quantity $$\begin{aligned}
\label{H}
\begin{split}
H&\equiv\left(ln\,R\right)^{.}\\
&=\frac{1}{3}\left(\frac{\dot{R_1}}{R_1}+\frac{\dot{R_2}}{R_2}+\frac{\dot{R_3}}{R_3}\right).
\end{split}\end{aligned}$$ where the dot denotes the time derivative. In order to have a compatible system, we set up the Einstein equations in a comoving frame with the matter source in which $u^{0}=1$ and $u^{i}=0$, $i=1,2,3$.\
In this frame, $p'=p-\lambda\dot{F}-3 \lambda F H$ and the stress-energy tensor components are $$\begin{aligned}
\label{Tmunu_comp}
\begin{split}
T^0_0&=-\rho, \\
T^i_i&=p'-2\eta F \frac{\dot{R_i}}{R_i}.
\end{split}\end{aligned}$$ Thus, the Einstein equations for the Bianchi I spacetime, having the viscous Lichnerowicz tensor as source (here only the $ii$ and $00$-components are non-vanishing), can be written as $$\begin{aligned}
\frac{\ddot{R_2}}{R_2}+\frac{\ddot{R_3}}{R_3}+\frac{\dot{R_2}\dot{R_3}}{R_2 R_3}&=-\chi\left(p'-2\eta F\frac{\dot{R_1}}{R_1}\right), \label{Einst1}  \\ 
\frac{\ddot{R_1}}{R_1}+\frac{\ddot{R_3}}{R_3}+\frac{\dot{R_1}\dot{R_3}}{R_1 R_3}&=-\chi\left(p'-2\eta F\frac{\dot{R_2}}{R_2}\right), \label{Einst2}  \\ 
\frac{\ddot{R_1}}{R_1}+\frac{\ddot{R_2}}{R_2}+\frac{\dot{R_1}\dot{R_2}}{R_1 R_2}&=-\chi\left(p'-2\eta F\frac{\dot{R_3}}{R_3}\right), \label{Einst3} \\ 
\frac{\dot{R_1}\dot{R_2}}{R_1 R_2}+\frac{\dot{R_1}\dot{R_3}}{R_1 R_3}+\frac{\dot{R_2}\dot{R_3}}{R_2 R_3}&=\chi\rho, \label{Einst4} \end{aligned}$$ $\chi$ being the Einstein constant.\
From the spatial components (\[Einst1\],\[Einst2\],\[Einst3\]), 