Observations indicate that on large scales ($\gg 100$ Mpc) the universe is homogeneous and isotropic and well described by spatially-flat FRW cosmologies. However, there is no reason to expect that such features should hold at the early stages of the evolution of the universe. Rather, it is generally accepted that anisotropy could have played a significant role in the early universe and that it has been fading away in the course of cosmic evolution. In the framework of 4-dimensional spacetime, a prototype for anisotropic vacuum cosmologies is provided by the Kasner metric [@Kasner], which mimics the behavior of more general solutions near the singularity during some finite periods of time[^3]. Various higher dimensional extensions of the vacuum Kasner model have been discussed in the literature [@Kokarev], [@Paul].

In this paper we consider spatially homogeneous but anisotropic cosmological models whose metric in $5D$ has the form $$\label{light-like 5D metrics}
dS^2 = {\cal{A}}(u, v)du dv - {\cal{B}}_{i j}(u, v)dx^{i}dx^{j},$$ where ${\cal{A}}$ and ${\cal{B}}$ are some functions of the “light-like" coordinates $u$ and $v$. These metrics are different from (\[Usual cosmological metric in 5D\]) in various aspects: (i) they do not contain the time or extra dimension in an explicit way; (ii) the 