= n_{y}\rho^{(eff)}, \;\;\;p_{z}^{(eff)} = n_{z}\rho^{(eff)},$$ where $n_{x}, n_{y}$ and $n_{z}$ are constants satisfying $n_{x} + n_{y} + n_{z} = 1$. If we average over the three spatial directions, this is equivalent to saying that the equation of state is ${\bar{p}}^{(eff)} = \rho^{(eff)}/3$, where ${\bar{p}}^{(eff)} \equiv - T_{i}^{i}/3$.

Barotropic linear equation of state 
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For the sake of generality, and in order to keep contact with isotropic FRW cosmologies, let us study the scenario where the effective mater is barotropic, that is the ratio $\bar{p}^{(eff)}/\rho^{(eff)}$ is constant. Thus we set $$\label{barotropic equation of state}
\bar{p}^{(eff)} = n \rho^{(eff)},\;\;\;n = \mbox{constant},$$ which for $n = 1$ and $n = 1/3$ gets back the above-discussed perfect fluid and radiation-like scenarios, respectively. Substituting into (\[EMT\]) we obtain an equation similar to (\[perfect fluid case\]) and (\[ultra-relativistic matter\]), but with the coefficient $(1 + 3n)$ in front of the term $c\dot{F}/F$. Consequently, $\dot{V} \propto \dot{F} F^{[c(1 + 3n)/(a + b)]}$. The condition $(\partial \dot{V}/\partial \psi) = (\partial {V'}/\partial t)$ then requires $$\label{F, barotropic case}
F = \left[f(t) + g(\psi)\right]^{\frac{(a + b)}{a + b + (3n + 1)c}}.$$ Substituting this expression into (\[Kasner-like metric in terms of F-bar\]) and making the coordinate transformation $(df/dt)dt \rightarrow d\tilde{t}$, 