Vector bundles and the icosahedron

A plane curve C defined by a homogeneous polynomial satisfying Laplace's equation appears canonically as the vanishing of the Pfaffian of a skew-symmetric matrix of linear forms. As a consequence there is a natural semi-stable rank two vector bundle defined on C. We consider the case of degree 3, and apply Atiyah's classification of bundles to determine various invariant strata in the space of harmonic cubics. We encounter the Mukai-Umemura threefold and link up with the classical geometry of the Clebsch diagonal cubic surface, certain distinguished rational curves on it, and the action of the icosahedral group.