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abstract: 'We consider spatially homogeneous, anisotropic cosmological models in $5D$ whose line element can be written as $dS^2 = {\cal{A}}(u, v)du dv - {\cal{B}}_{i j}(u, v)dx^{i}dx^{j}$, $(i, j = 1, 2, 3)$, where $u$ and $v$ are light-like coordinates. In 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)$. Among them, there is a family of self-similar homothetic solutions that contains, as a particular case, the so-called light-like Kasner universes. In this work we provide a detailed study of the different types of $4D$ scenarios that can be embedded in such universes. For the sake of generality of the discussion, and applicability of the results, in our analysis we consider the two versions of non-compactified $5D$ relativity in vogue, viz., braneworld theory and induced matter theory. We find a great variety of cosmological models in $4D$ which are anisotropic versions of the FRW ones. We obtain models on the brane with a non-vanishing cosmological term $\Lambda_{(4)}$, which inflate [*à la*]{} de Sitter without satisfying the classical false-vacuum equation of state. Using the symmetry of the solutions, 