Extended Il'Yashenko category conjecture for non-integrable SR spheres

Let an SR sphere mean a sphere for a sub-Riemannian structure, and call the case non-integrable when the associated geodesic dynamics is not integrable. Il'Yashenko's category is the class of functions described by super-accurate expansions, including flat exponential terms; an extended Il'Yashenko category is a broader category of this type that also accommodates parameters. Extended Il'Yashenko category conjecture. In the general non-integrable case, SR spheres belong to some extended Il'Yashenko category. The proposed category is intended to capture the parameter-dependent Poincaré return mappings arising from the pendulum representation, but its construction and the conjectured membership remain open.