  Let X be a complex-projective contact manifold whose second Betti-number is
one. It has long been conjectured that X should then be rational-homogeneous,
or equivalently, that there exists an embedding of X into a projective space
whose image contains lines.
  Using methods introduced in math.AG/0206193, we show that X is covered by a
compact family of rational curves, called "contact lines" that behave very much
like the lines on the rational homogeneous examples: if x in X is a general
point, then all contact lines through x are smooth, no two of them share a
common tangent direction at x, and the union of all contact lines through x
forms a cone over an irreducible, smooth base. As a corollary, we obtain that
the tangent bundle of X is stable.
