Kahler metrics whose geodesics are circles

We classify all Kahler metrics in an open subset of $C^2$ whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to Fubini-Study metric on $C{P}^2$ restricted to an affine chart, to the complex hyperbolic metric in the unit ball model or to the Euclidean metric).