\rightarrow d\bar{u} $, it reduces to $$\label{Lightlike Kasner solution}
dS^2 = d\bar{u} dv -A\bar{u}^{p_{1}}dx^2 - B \bar{u}^{p_{2}}dy^2 - C \bar{u}^{p_{3}}dz^2,$$ where $A, B, C$ are constants with the appropriate units, and $p_{1},  p_{2},  p_{3}$ denote $$\label{introduction of p1, p2 and p3}
p_{1} = \frac{2 \alpha (\alpha + \beta + \gamma)}{\alpha^2 + \beta^2 + \gamma^2}, \;\;\;p_{2} = \frac{2 \beta (\alpha + \beta + \gamma)}{\alpha^2 + \beta^2 + \gamma^2}, \;\;\;p_{3} = \frac{2 \gamma (\alpha + \beta + \gamma)}{\alpha^2 + \beta^2 + \gamma^2},$$ which satisfy the relation $\sum_{i = 1}^{3}\left(p_{i} - 1\right)^2 = 3$ for [*any*]{} values of $\alpha, \beta $ and $\gamma$. The metric (\[Lightlike Kasner solution\]) is usually called light-like Kasner solution. In this case the $5D$ homothetic vector is given by ${\zeta}^C = \left[\bar{u},\;  v,\;  (1 - p_{1}/2) x,\;  (1 - p_{2}/2) y,\;  (1 - p_{3}/2) z\right]$.

Introducing the timelike and “extra" coordinates
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In order to be able to apply the standard dimensional reduction (\[4D Einstein with T and K\]) to metrics (\[light-like 5D metrics\]) one has to introduce coordinates that are adapted to the spacetime sections $\Sigma_{\psi}$. With this aim we make the coordinate transformation

$$\label{definition of F and V}
u = F(t, \psi), \;\;\;v = V(t, \psi),$$

