  We discuss the $\beta$-deformation of $AdS_5\times S^5$ which incorporates
the $SL(2,\mathbb{R})$ symmetry of the type IIB theory. The axion-dilaton is
identified with a two-torus from an eleven dimensional viewpoint. We consider
the null geodesic with equal component angular momenta to take the Penrose
limit of the deformed $AdS_5\times S^5$. We study the bosonic part of the
string sigma model and the spectrum of the string in the pp-wave background.
