Superisometries and integrability of superstrings

For type II supergravity backgrounds with superisometries the corresponding transformations and conserved currents for the superstring are constructed up to fourth order in $\Theta$. It is then shown how, for certain backgrounds related to near horizon geometries of intersecting branes, the components of the superisometry current can be used to construct a Lax connection demonstrating the classical integrability of the string in these backgrounds. This includes examples of $AdS_2$ and $AdS_3$ backgrounds with a $D(2,1;\alpha)$ isometry group which have not previously been studied from an integrability point of view. The construction of the Lax connection is carried out up to second order in $\Theta$.