0, \nonumber \\
a^2 \left[c_{1} u + c_{2}v\right]{\cal{A}}_{v} &+& 2 c \; c_{2} {\cal{A}} = 0,\end{aligned}$$ which have a unique solution given by ${\cal{A}} = C_{0} \left(c_{1} u + c_{2} v\right)^{- 2c/a^2}$, where $C_{0}$ is a constant of integration. Now, it is easy to verify that $R_{u v} = 0$ is identically satisfied. In summary, the final form of the self-similar solution is given by[^5] $$\label{new solution}
{\cal{A}} = \left(c_{1} u + c_{2} v\right)^{- 2c/a^2}, \;\;\;{\cal{B}}_{11} =  {\cal{A}}^{- \alpha a/c}, \;\;\;{\cal{B}}_{22} =  {\cal{A}}^{- \beta a/c}, \;\;\;{\cal{B}}_{33} =  {\cal{A}}^{- \gamma a/c}, \;\;\;{\cal{B}}_{i j} = 0, \;\;\;i \neq j.$$

We note that this solution admits a homothetic Killing vector in $5D$ for any values of $\alpha$, $\beta$ and $\gamma$, namely, $$\label{Lie derivative of the self-similar metric}
{\cal{L}}_{{\zeta}}{g_{A B}} = 2 g_{A B}, \;\;\;\mbox{with}\;\;\;    {\zeta}^C = \left[ {\eta}_{0}  u,\;  {\eta}_{0}  v,\;  (1 - \alpha {\eta}_{0}/a) x,\;  (1 - \beta {\eta}_{0}/a) y,\;  (1 - \gamma {\eta}_{0}/a) z \right],$$ where $g_{A B}$ is the metric (\[new solution\]), ${\cal{L}}_{\zeta}$ denotes the Lie derivative along the $5D$ vector ${\zeta}^C$ and ${\eta}_{0} \equiv a^2/(a^2 - c)$. In addition, by setting one of the constants equal to zero, say $c_{2} = 0$, and making the coordinate transformation $u^{- 2c/a^2}du 