are defined as $$\begin{array}{lcl}
        z^1  = R \varphi_1 \,  , & \phantom{qquad} & z^2  =
        \displaystyle{ \frac{R}{\sqrt{2}}} \left(\displaystyle{\frac{ \pi}{4}-\psi}\right) \,  , \\[4mm]
        z^3 = R \varphi_2 \,  , &  & z^4  = R \left(\displaystyle{\frac{ \pi}{4}}-\theta \right) \,  .
    \end{array}$$ while $z^5$, $z^6$, $z^7$, $z^8$ are again given by Eq.. More explicitly we have $$\begin{aligned}
    r_3  = R \rho \sin \beta \, , \qquad     r_4  = R \rho \cos \beta\,  ,\end{aligned}$$ $$\begin{aligned}
    z^5 + i z^6  = r_3 \displaystyle{ e^{i \varphi_3}} \, , \qquad z^7  + i z^8 = r_4 \displaystyle{ e^{i \varphi_4}}\,  .\end{aligned}$$ Substituting the new coordinates in the background – and taking the Penrose limit we get the following pp-wave metric $$\label{eq:dstwoflat}
    \begin{split}
        ds^2&=-4dz^+dz^- + dz^i dz^i - \mu^2 \sum_{k=3,4}
        \left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\left(dz^+\right)^2 \\
        &+ 2\mu \sum_{k=3,4}\eta_k\left[ z^{2k-1}dz^{2k}-  z^{2k}dz^{2k-1}\right]dz^+
        - 4\mu\left(z^2 dz^1 + z^4 dz^3\right)dz^+.
    \end{split}$$ and the five-form is defined in . This is a new pp-wave background and it has two explicit isometries, $z^1$ and $z^3$ We will therefore refer to it as [*two flat directions pp-wave background*]{}.

In this case $\varphi_3,\varphi_4$ are fixed, thus, keeping in mind , we have that also the angular momenta $S_1$ and $S_2$ are fixed. In a similar fashion as before if we 