means that $J_3$, $S_1$ and $S_2$ are fixed. But now the condition that $H_\textrm{lc}$, $p^+$ and $p^1$ have to remain finite in the limit tells us that the quantities $$E-J_1-J_2 , \quad \frac{E+J_1+J_2}{R^2}, \quad \frac{J_1+J_2}{R^2} , \quad \frac{J_1-J_2}{R} , \quad g_s,l_s$$ are all fixed when $R \to \infty$. This is the regime corresponding to the “one flat direction” Penrose limit of type IIB string theory on $\mbox{AdS}_5\times S^5$, as found in [@Bertolini:2002nr]. Translating this into ${\mathcal{N}}=4$ SYM language, it corresponds to the regime where [@Bertolini:2002nr] $$E-J_1-J_2 , \quad \frac{E+J_1+J_2}{\sqrt{N}}, \quad \frac{J_1+J_2}{\sqrt{N}} , \quad \frac{J_1-J_2}{N^{1/4}} , \quad {g_{\rm YM}}^2$$ are fixed for $N \to \infty$.

The “two flat directions” Penrose limit
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We finally consider the Penrose limit that leads to a new pp-wave  with two flat directions. The variables $\varphi_0,$ $\varphi_1,$ $\varphi_2,$ $\varphi_3,$ $\varphi_4$ are now defined as $$\begin{gathered}
\label{phi2fd}
        \chi  = \varphi_0 - \sqrt{2}\varphi_1 - \varphi_2 \,  , \qquad
        \phi  = \varphi_0 + \sqrt{2}\varphi_1 - \varphi_2\,  , \qquad
        \alpha  = \varphi_0 + \varphi_2 \,  ,{\nonumber}\\[2mm]
        \gamma  = \eta_3 \varphi_0 + \varphi_3  \, , \qquad
        \xi  =  \eta_4 \varphi_0 + \varphi_4 \, ,\end{gathered}$$ whereas the light-cone coordinate are as usual given by .

The coordinates $z^1$, $z^2$, $z^3$ and $z^4$ 