Geodesic Flow on Global Holomorphic Sections of TS^2

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 ${\mathbb{R}}^3$, 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.