of solutions are presented in the Appendix) and introduce a timelike coordinate $t$ and a spacelike coordinate $\psi$ along the extra dimension. This is equivalent to introducing two additional degrees of freedom, which are expressed in terms of two functions of $t$ and $\psi$. We will see that, as in the familiar FRW picture (\[Usual cosmological metric in 5D\]), these two functions can be related to the specific choice for embedding $\Sigma_{\psi}$ in $5D$ and to the physics in $4D$. In section $3$, within the context of STM we show that the light-like Kasner solutions generate a great variety of cosmological models in $4D$, including the de Sitter, Milne and power-law FRW models. In section $4$, within the context of the braneworld paradigm we find that they embed $4D$ cosmological models with a non-vanishing cosmological term $\Lambda_{(4)}$, which in principle can be either constant or time-dependent. In the case of constant $\Lambda_{(4)}$ the $3D$ space exponentially inflates, regardless of the specific embedding. We also show that by virtue of the symmetry $(x, y, z) \leftrightarrow \psi$, they generate a class of non-static vacuum solutions on the brane. In section $5$, once again using symmetry properties, we demonstrate that the Kasner-like 