hypersurfaces of constant $u$ or $v$ are three-dimensional instead of $4D$; (ii) [*[a priori]{}*]{} it is not clear how to define the $5D$ unit vector ${\hat{n}}^{A}$ along the extra dimension, which in turn is needed for defining the spacetime sections and for constructing the appropriate projected quantities in $4D$.

Here, for the case where ${\cal{B}}_{i j}$ is diagonal we construct three families of analytic solutions to the $5D$ vacuum field equations $R_{AB} = 0$ $ (A, B = 0, 1, 2, 3, 4)$. The simplest one is a family of self-similar solutions[^4], which contains as a particular case the so-called light-like Kasner universes. From a physical point of view, self-similar homothetic models are interesting because they may serve as asymptotic regimes, i.e., near the initial cosmological singularity and at late times, for many homogeneous and inhomogeneous cosmological models [@Coley]. The other two families of solutions are obtained under the assumption that some of the metric coefficients are separable functions of their arguments.

In view of their potential relevance to the “similarity hypothesis" [@Coley], in this work we focus our attention to the family of self-similar $5D$ spacetimes mentioned above. The main question under study is what kind of $4D$ scenarios 