Remarks on the intersection of two quadrics

The article takes the formula for the integrable system defined by Beauville et al on the cotangent bundle of the intersection of two quadrics X, and interprets it in terms of rank 2 quasi parabolic Higgs bundles on the projective line. We then discuss aspects related to the geometric Langlands programme in this simple concrete context. We conclude with a description of the link with the recent paper of Benedetti et al identifying X and its integrable system in terms of invariant Spin(2g) bundles on a hyperelliptic curve of genus g.