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Twistor CR manifolds, introduced by LeBrun, are Lorentzian (neutral) CR5-manifolds defined as $\mathbb{P}^1$-bundles over 3-dimensional conformalmanifolds. In this paper, we embed a real analytic twistor CR manifold into thetwistor space of the anti self-dual Poincar\'e-Einstein metric whose conformalinfinity is the base conformal 3-manifold, and construct the associatedFefferman ambient metric as a neutral hyperk\"ahler metric on the spinor bundlewith the zero section removed. We also describe the structure of the Cheng--Yautype K\"ahler-Einstein metric which has the twistor CR manifold as the boundaryat infinity.

Hyperk\"ahler ambient metrics associated with twistor CR manifolds