  We describe the range of the Radon transform on the space $M$ of irreducible
conics in $\CP^2$ in terms of natural differential operators associated to the
$SO(3)$-structure on $M=SL(3, \R)/SO(3)$ and its complexification. Following
\cite{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 ${\mathcal Z}$ admits a holomorphic fibration over $\CP^2$. In
the special case where ${\mathcal Z}=\CP^3\setminus\CP^1$ the twistor lines
project down to a four-parameter family of conics which form triangular
Poncelet pairs with a fixed base conic.
