Integrability of all one-phase cones with isolated singularities

Determine whether every one-phase cone with an isolated singularity is integrable.

Background

A one-phase cone with an isolated singularity is a nontrivial 1-homogeneous global solution whose only possible singular point is the origin. The paper defines integrability by requiring that every 1-homogeneous Jacobi field arise as the infinitesimal variation of a family of 1-homogeneous classical one-phase solutions.

The paper emphasizes that its uniqueness theorem does not require this structural assumption: uniqueness of tangent cones is proved even for potentially non-integrable cones. Nevertheless, whether integrability is universal among cones with isolated singularities remains unresolved, and a general classification of singular one-phase cones is also unavailable.