S-matrix as assumed for the Bethe ansatz in [@Staudacher:2004tk], and the same anomalous dimensions as computed using this Bethe ansatz in [@Staudacher:2004tk]. A complementary approach to confirming this Bethe ansatz will appear in [@Eden:2005xx]. Evidence for two-loop integrability in the fermionic ${{\mathfrak{sl}}}(2)$ sector was given in [@Belitsky:2005bu].

Section \[sec:psu112\] introduces the $ {{\mathfrak{su}}}(1,1|2)$ sector and the residual symmetry algebra. Section \[sec:g2\] discusses the $\mathcal{O}(g^2)$ solution, and Section \[sec:g3\] discusses the $\mathcal{O}(g^3)$ solution and presents the two-loop dilatation generator. For simplicity, we assume planarity until Section \[sec:non-planar\], and in that section we present the lift to the finite-$N$ solution. In Section \[sec:checks\], after verifying that our solution predicts the same anomalous dimensions as those of the field theory calculations of [@Beisert:2003tq; @Kotikov:2003fb; @Eden:2005bt; @Belitsky:2005bu], we summarize the applications of the solution to compute the bosonic ${{\mathfrak{sl}}}(2)$ subsector S-matrix and some anomalous dimensions. We conclude and discuss directions for further research in Section \[sec:outlook\]. The appendix presents details about the symmetry algebra and proofs of our solution. Finally, we use many results and notations of [@Beisert:2004ry].

The $ {{\mathfrak{su}}}(1,1|2)$ sector \[sec:psu112\]
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As explained in [@Beisert:2004ry], it is consistent to restrict to various sectors of the states of $\mathcal{N}=4$ SYM. Under such 