Emptiness of the regular component in fibers over rank-1 critical values for p=2

Ascertain whether, for p=2 and any rank-1 critical value (j,h) of the p-adic Jaynes–Cummings model F=(J,H): S^2_p × (Q_p)^2 → (Q_p)^2, the fiber F^{-1}({(j,h)}) consists solely of the circle of rank-1 critical points, or whether there exists a nonempty 2-dimensional p-adic analytic submanifold component of the fiber.

Background

For p=2, Theorem 6.3 states that a fiber over a rank-1 critical value is the disjoint union of a circle of critical points and possibly an additional 2-dimensional analytic submanifold, which may be empty. The figures suggest the regular part might be empty, but the authors cannot derive this from their formulas.

Resolving this question would clarify the topology of fibers at rank-1 critical values in the 2-adic case and determine whether noncritical points occur in such fibers.