+ \lambda_{u}\left( \sigma_{v} + 2 \lambda_{v} + \mu_{v}\right) + \lambda_{v}\left(\mu_{u} + \sigma_{u}\right) &=& 0, \nonumber \\
4 \mu_{u v} + \mu_{u}\left( \lambda_{v} + 2 \mu_{v} + \sigma_{v}\right) + \mu_{v}\left(\sigma_{u} + \lambda_{u}\right) &=& 0, \nonumber \\
4 \sigma_{u v} + \sigma_{u}\left( \mu_{v} + 2 \sigma_{v} + \lambda_{v}\right) + \sigma_{v}\left(\lambda_{u} + \mu_{u}\right) &=& 0.\end{aligned}$$ Here the subscripts $u$, $v$ indicate partial derivatives with respect to those arguments. The above equations show cyclic permutation symmetry, i.e., starting from any of them by means of the transformation $\lambda \rightarrow \mu \rightarrow \sigma \rightarrow \lambda$ we obtain the other two.

$\bullet$ Self-similar solutions: First we solve the field equations under the assumption that the metric (\[light-like 5D metrics\]) possesses self-similar symmetry. This assumption is motivated by a number of studies suggesting that self-similar models play a significant role at asymptotic regimes [@Coley]. From a mathematical point of view, it means that by a suitable transformation of coordinates all the dimensionless quantities can be put in a form where they are functions only of a single variable (say $\zeta$) [@Sedov]. In our particular case, this implies that $\lambda = \lambda(\zeta)$, $\mu = \mu(\zeta)$, $\sigma = \sigma(\zeta)$, where $\zeta$ is some function of $u$ and $v$, viz., 