denote $$\label{notation}
a \equiv \alpha + \beta + \gamma, \;\;\;b \equiv \alpha^2 + \beta^2 + \gamma^2  - \alpha - \beta - \gamma, \;\;\;c \equiv \alpha \beta + \alpha\gamma + \beta \gamma,$$ where $\alpha, \beta, \gamma$ are arbitrary parameters.

A simple integration gives $$\label{f(u, v)}
f = \left[h(u) + g(v)\right]^{1/a},$$ where $h(u)$ and $g(v)$ are arbitrary functions of their arguments. Clearly, in the present case ${\cal{B}}_{i j}$ are power-law type solutions of the similarity variable $\zeta = \left[h(u) + g(v)\right]$.

To simplify the discussion, and eliminate spurious degrees of freedom, we now make the coordinate transformation $h(u) = c_{1} \bar{u}$, $g(v) = c_{2} \bar{v}$, where $c_{1}$ and $c_{2}$ are constants. In these new coordinates $${\cal{A}}(u, v)du dv \rightarrow \bar{{\cal{A}}}(\bar{u}, \bar{v})d\bar{u} d\bar{v},$$ where $\bar{{\cal{A}}}(\bar{u}, \bar{v}) = \left[{\cal{A}}(u, v)/h_{u}g_{v}\right]$ with $u$ and $v$ expressed in terms of $\bar{u}$, $\bar{v}$. Then relabeling the coordinates (dropping the overbars) the metric becomes

$$dS^2 = {\cal{A}}(u, v) du d v - C_{1}\left[c_{1} u + c_{2}v\right]^{2\alpha/a}dx^2 - C_{2}\left[c_{1} u + c_{2}v\right]^{2\beta/a}dy^2 - C_{3}\left[c_{1} u + c_{2}v\right]^{2\gamma/a}dz^2.$$

For this metric the field equations $R_{u u} = 0$ and $R_{vv} = 0$ yield the equations $$\begin{aligned}
\label{Ruu, Rvv}
a^2 \left[c_{1} u + c_{2}v\right]{\cal{A}}_{u} &+& 2 c \; c_{1} {\cal{A}} = 