Reduction of beta-integrable 2-Segre structures

We show that locally every beta-integrable (2,n)-Segre structure can be reduced to a torsion-free S^1*GL(n,R)-structure. This is done by observing that such reductions correspond to sections with holomorphic image of a certain `twistor bundle’. For the homogeneous (2,n)-Segre structure on the oriented 2-plane Grassmannian, the reductions are shown to be in one-to-one correspondence with the smooth quadrics in CP^{n+1} without real points.