compute $H_\textrm{lc}$, $p^+$, $p^1$ and $p^3$ and request that they should stay finite in the Penrose limit we get that the quantities $$E-J_1-J_2-J_3 , \quad \frac{E+J_1+J_2+J_3}{R^2},
\quad
\frac{J_1+J_2+J_3}{R^2}, \quad \frac{J_1 - J_2}{R} ,\quad  \frac{J_3 -J_1 - J_2}{R}, \quad g_s,l_s$$ are fixed as $R$ goes to infinity. This is the regime corresponding to the “two flat directions” Penrose limit of type IIB string theory on $\mbox{AdS}_5\times S^5$. Translating this into ${\mathcal{N}}=4$ SYM it corresponds to the regime where $$E-J_1-J_2-J_3 , \quad \frac{E+J_1+J_2+J_3}{\sqrt{N}}, \quad
\frac{J_1+J_2+J_3}{\sqrt{N}}, \quad \frac{J_1 - J_2}{N^{1/4}} ,\quad  \frac{J_3 -J_1 - J_2}{N^{1/4}} , \quad {g_{\rm YM}}^2$$ are fixed for $N \rightarrow \infty$. Here $J_1-J_2$ and $J_3-J_1-J_2$ correspond to the two momenta for the two space-like isometries of the [*two flat directions pp-wave background*]{} .

Type IIB string theory on the pp-wave backgrounds , (with five-form field strength given by ) can be easily quantized. The spectra in all these three cases are worked out in the next section.

String theory spectrum on a rotated pp-wave background {#sec:stringrotspectra}
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In this section we obtain a pp-wave metric, which depends on parameters introduced through a coordinate transformation on the maximally   of [@Blau:2001ne]. For this reason, in practice, this metric describes an infinite 