  We study the geodesic flow on the global holomorphic sections of the bundle
$\pi:{TS}^2\to {S}^2$ induced by the neutral K\"ahler metric on the space of
oriented lines of ${\Bbb{R}}^3$, which we identify with ${TS}^2$. This flow is
shown to be completely integrable when the sections are symplectic and the
behaviour of the geodesics is described.
