metric (\[Lightlike Kasner solution\]) can be used to generate static pancake-like distributions of matter in $4D$, similar to those proposed by Zel’dovich for the shape of the first collapsed objects in an expanding anisotropic universe [@Zel'dovich]. In section $6$ we present a summary of our results.

Cosmological models in $5D$. Light-like coordinates
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Solving the field equations. Part I
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In this work we obtain three families of solutions to the field equations $R_{A B} = 0$. However, to facilitate the discussion, in this subsection we present only one of them. Specifically, we present the family of solutions that we will use throughout the paper, which (as we will see in sections $3$-$5$) may be interpreted or used as $5D$ embeddings for a number of $4D$ universes. The derivation of the other two families of solutions, whose $4D$ interpretation is not discussed here, is deferred to the Appendix (Solving the field equations. Part II).

To simplify the shape of the field equations let us momentarily denote ${\cal{B}}_{11} = e^{\lambda(u, v)}$, ${\cal{B}}_{22} = e^{\mu(u, v)}$ and ${\cal{B}}_{33} = e^{\sigma(u, v)}$. From $R_{xx} = 0$, $R_{yy} = 0$ $R_{zz} = 0$ we obtain the equations $$\begin{aligned}
\label{Rxx, Ryy, Rzz}
4 \lambda_{u v} 