In this chapter, the author describes his collaboration with Lionel Mason and Roman Maszczyk on the relationship between two central ideas in the theory of integrable systems. The author builds on ideas of Nigel Hitchin and suggests the possibility of new applications of twistor theory to problems in classical analysis. The first idea is an old one where the prominence is given by Sofya Kowalevskaya. The second idea is more recent where there is a fundamental connection between self-duality and integrability. The first connection with the self-duality equations comes from the correspondence between isomonodromic families of ODEs with four poles and holomorphic bundles over a neighbourhood of a line in twistor space invariant under a three-dimensional abelian Lie algebra g of infinitesimal projective transformations. The chapter concludes that the bundles on twistor space are defined only on the neighbourhood of a line.