  This article considers the attenuated transport equation on Riemannian
surfaces in the light of a novel twistor correspondence under which matrix
attenuations correspond to holomorphic vector bundles on a complex surface. The
main result is a transport version of the classical Oka-Grauert principle and
states that the twistor space of a simple surface supports no nontrivial
holomorphic vector bundles. This solves an open problem on the existence of
matrix holomorphic integrating factors on simple surfaces and is applied to
give a range characterisation for the non-Abelian X-ray transform. The main
theorem is proved using the inverse function theorem of Nash and Moser and the
required tame estimates are obtained from recent results on the injectivity of
attenuated X-ray transforms and microlocal analysis of the associated normal
operators.
