Twistor reductions of the KdV hierarchy

It is known that the KdV hierarchy admits symmetry reductions that give rise to hierarchies of integrable ODE. In the simplest case of the KdV equation, such reductions give rise to the Painlevé I and Painlevé II equations. In [L.J. Mason, N.M.J. Woodhouse, Integrability, Self-Duality, and Twistor Theory, in: London Mathematical Society Monographs New Series, vol. 15, Oxford University Press, Oxford, 1996], a twistor description of these similarity reductions was constructed in terms of invariant vector bundles in the twistor space. In this paper we generalize this construction to the case of the KdV hierarchy to obtain the twistor spaces of the Painlevé I and Painlevé II hierarchy as twistor reductions of the KdV hierarchy.