Twistor theory of the Chen--Teo gravitational instanton

Toric Ricci--flat metrics in dimension four correspond to certain holomorphic vector bundles over a twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, for the five--parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen--Teo family contains a two--parameter family of asymptotically flat gravitational instantons. The patching matrices for these instantons take a simple rational form.