Recently, Kallen and Zabzine computed the partition function of a twisted
supersymmetric Yang-Mills theory on the five-dimensional sphere using
localisation techniques. Key to their construction is a five-dimensional
generalisation of the instanton equation to which they refer as the contact
instanton equation. Subject of this article is the twistor construction of this
equation when formulated on K-contact manifolds and the discussion of its
integrability properties. We also present certain extensions to higher
dimensions and supersymmetric generalisations.Comment: v3: 28 pages, clarifications and references added, version to appear
in JHE
