  This paper presents a new perspective on integrability in theories of
gravity. We show how the stationary, axisymmetric sector of General Relativity
can be described by the boundary dynamics of a four-dimensional Chern-Simons
theory. This approach shows promise for simplifying solution generating methods
in both General Relativity and higher-dimensional supergravity theories. The
four-dimensional Chern-Simons theory presented generalises those for flat space
integrable models by introducing a space-time dependent branch cut in the
spectral plane. We also make contact with twistor space approaches to
integrability, showing how the branch cut defects of four-dimensional
Chern-Simons theory arise from a discrete reduction of six-dimensional
Chern-Simons theory.
