  Using the twistor correspondence, this article gives a one-to-one
correspondence between germs of toric anti-self-dual conformal classes and
certain holomorphic data determined by the induced action on twistor space.
Recovering the metric from the holomorphic data leads to the classical problem
of prescribing the Cech coboundary of 0-cochains on an elliptic curve covered
by two annuli. The classes admitting Kahler representatives are described; each
such class contains a circle of Kahler metrics. This gives new local examples
of scalar-flat Kahler surfaces and generalises work of Joyce who considered the
case where the distribution orthogonal to the torus action is integrable.
