can be embedded in such spacetimes. For the sake of generality of the discussion, and applicability of our results, in our analysis we consider both versions of non-compactified $5D$ relativity, viz., induced-matter and brane theory. Unfortunately, the expressions obtained in $4D$ as projections of the $5D$ solutions are quite complex and cumbersome. Therefore, to obtain manageable mathematical expressions in $4D$, in sections $3$, $4$ and $5$ we simplify the algebra (but not the physics) by restricting our discussion to the subset of light-like Kasner solutions.

Our solutions generalize a number of isotropic cosmological models and give back previous ones in the literature (see e.g., [@JPdelW]-[@JPdeL-JCAP] and references therein). Although we are not discussing particular applications here, they could be useful in the study of generalizations of Mixmaster or Belinskii-Khalatnikov-Lifshitz oscillations in theories with a single extra dimension [@mixmaster], [@BKL], [@Halpern]-[@Henneaux]. They could also be applied to studying conjectures about isotropic Big Bang singularities in braneworlds [@Maartens]-[@Coley2]. Certain cosmological models, such as the cyclic universe model, also require an understanding of the behavior of Kasner-like solutions and are based on brane-type models [@Erickson].

The paper is organized as follows. In section $2$ we derive our self-similar solution (the other two families 