Configurations of lines on contact Fano manifolds

I will address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. One of the main tools to approach this problem are minimal rational curves, which are called contact lines in this case. I will investigate possible configurations of contact lines and their deformations. In this way we can prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from the geometry of an arbitrary contact Fano manifold.