Exceptional geometry is an attempt to combine the geometric diffeomorphisms and matter gauge transformations in gravity-matter theories into a single geometric structure. I will review recent results associated with a 2+9 split of maximal supergravity where the affine symmetry group E9 plays a central role. The results also provide a general formula that is applicable to many other cases.

I will review the known integrable structures of (super-)gravity in the presence of a sufficient number of Killing vectors and how they can be used to generate new solutions. The associated inverse scattering techniques have been widely applied to four-dimensional Einstein gravity, starting with the work of Belinski and Zakharov, but their generalisation to other gravity systems poses some problems. I will discuss progress for the case of STU supergravity by exploiting the underlying group theory.