Self-associated three-dimensional cones

For every proper convex cone $K \subset \mathbb R^3$ there exists a unique complete hyperbolic affine 2-sphere with mean curvature $-1$ which is asymptotic to the boundary of the cone. Two cones are associated if the corresponding affine spheres can be mapped to each other by an orientation-preserving isometry. This equivalence relation is generated by the groups $SL(3,\mathbb R)$ and $S^1$, where the former acts by linear transformations of the ambient space, and the latter by multiplication of the cubic holomorphic differential of the affine sphere by unimodular complex constants. The action of $S^1$ generalizes conic duality, which acts by multiplication of the cubic differential by $-1$. We call a cone self-associated if it is linearly isomorphic to all its associated cones, in which case the action of $S^1$ induces (nonlinear) isometries of the corresponding affine sphere. We give a complete classification of the self-associated cones and compute isothermal parametrizations of the corresponding affine spheres. The solutions can be expressed in terms of degenerate Painlev\'e III transcendents. The boundaries of generic self-associated cones can be represented as conic hulls of vector-valued solutions of a certain third-order linear ordinary differential equation with periodic coefficients, but there exist also cones with polyhedral boundary parts.