way $$\begin{aligned}
\label{eq:oneflatphi}
\chi = \varphi_0 -\varphi_1\, , \quad
\phi = \varphi_0 +  \varphi_1\, , \quad
\alpha = \eta_2 \varphi_0 +  \varphi_2\, , \quad
\gamma = \eta_3\varphi_0 + \varphi_3\, , \quad
\xi = \eta_4\varphi_0 + \varphi_4\, ,\end{aligned}$$ with the light-cone variables still given by eq.n .

We moreover define $z^1$ and $z^2$ as $$\begin{aligned}
z^1 = R\varphi_1\, , \quad z^2=R\left(\frac{\pi}{4}-\psi\right)\, ,\end{aligned}$$ while $z^3,...,z^8$ are defined as before (see Eq.) and $$\begin{aligned}
r_2 = R \theta\, , \quad
r_3 = R \rho \sin\beta\, ,\quad
r_4= R \rho \cos\beta\, ,\end{aligned}$$ $$\begin{aligned}
z^3+iz^4 =r_2 e^{i\varphi_2}\, , \quad
z_5+iz_6 = r_3e^{i\varphi_3}\, , \quad z_7+iz_8 = r_4e^{i\varphi_4}\, .\end{aligned}$$ The Penrose limit is then the limit $R\to\infty$ keeping $z^\pm,z^i$ fixed. Plugging the coordinates $z^\pm, z^i$ into the background – and taking the limit described above the metric becomes $$\label{eq:dsoneflat}
    \begin{split}
        ds^2=&-4dz^+dz^- + dz^i dz^i - \mu^2 \sum_{k=2}^{4}
        \left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\left(dz^+\right)^2 \\
        &+ 2\mu \sum_{k=2}^{4}\eta_k \left[z^{2k-1}dz^{2k}- z^{2k}dz^{2k-1}\right]dz^+ -4 \mu z^2 dz^+ dz^1.
    \end{split}$$ with the five-form given by .

From we see that $z^1$ is an explicit isometry of the above pp-wave background and therefore we call this background [*one flat direction pp-wave background*]{}.

As before we have that $\varphi_2,\varphi_3,\varphi_4$ are fixed in the Penrose limit which, using , 