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We shall describe the twistor space of a hyperka¨hler 4n-manifold with an isometric S1-action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimension n.We deduce that any hyperka¨hler metric on the cotangent bundle of a real-analytic Ka¨hler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Ka¨hler metric and for which the S1-action by scalar multiplication in the fibres is isometric is unique in a neighbourhood of the zero section. These metrics have been constructed independently by the author and Kaledin. [Copyright &y& Elsevier]