Conics, twistors, and anti-self-dual tri-Kähler metrics

We describe the range of the Radon transform on the space M of irreducible conics in ² in terms of natural differential operators associated to the $SO(3)$-structure on M=SL(3, )/SO(3) and its complexification. Following {moraru} we show that for any function F in this range, the zero locus of F is a four-manifold admitting an anti-self-dual conformal structure which contains three different scalar-flat K\"ahler metrics. The corresponding twistor space Z admits a holomorphic fibration over ². In the special case where Z=³¹ the twistor lines project down to a four-parameter family of conics which form triangular Poncelet pairs with a fixed base conic.