In fact, in this case we can re-scale the function $F$, as $F \rightarrow \bar{F}^{a^2/(a + b)}$, after which the solution (\[light-like metric in t-psi coordinates\]) with $c_{2} = 0$ reduces to $$\label{Kasner-like metric in terms of F-bar}
dS^2 = \dot{\bar{F}}\dot{V} dt^2 - A {\bar{F}}^{p_{1}}dx^2 - B {\bar{F}}^{p_{2}}dy^2 - C {\bar{F}}^{p_{3}}dz^2 -       \frac{{\bar{F}}'^2 \dot{V}}{\dot{\bar{F}}} d\psi^2,$$ where $p_{1}, p_{2}, p_{3}$ are the parameters introduced in (\[introduction of p1, p2 and p3\]). In sections $3$, $4$ and $5$ we use this line element, which we call Kasner-like, for illustrating the fact that physics in $4D$ determines $F$.

Cosmological models in $4D$. The STM approach
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The aim of this section is to determine $F$ within the context of induced matter theory. To this end we assume an equation of state for the effective matter quantities. Our results show that the light-like Kasner metrics (\[Lightlike Kasner solution\]) can be used, or interpreted, as $5$-dimensional embeddings for a number of cosmological models in $4D$ that are spatially anisotropic extensions of the FRW ones.

For the Kasner-like metric (\[Kasner-like metric in terms of F-bar\]), the components of the effective EMT induced on $\Sigma_{\psi}: \psi = \psi_{0} = $ constant are given by (in what follows 