Determine when the singular integral affine base determines the integrable system

Determine precise conditions under which the singular integral affine manifold B, obtained as the base of the (generally singular) Lagrangian fibration induced by a smooth integrable system (M,ω,F), uniquely determines the integrable system up to fiberwise symplectomorphism. This includes making explicit the class of integrable systems for which a rigorous notion of singular integral affine structure on all of B can be defined and then establishing whether B, equipped with this structure, recovers (M,ω,F) up to the natural equivalence.

Background

In regular regions of an integrable system (M,ω,F), the Liouville–Arnold–Mineur theorem yields action–angle coordinates and induces an integral affine structure on the set of regular values B_r. In the toric case, the image polytope encodes this affine structure and classifies the system up to isomorphism.

Beyond the toric case, one would like to extend this to a singular integral affine structure on the full base B=M/∼, but fibers may be disconnected or singular, complicating the definition. The remark frames a central question: whether such a singular affine structure on B suffices to determine the system up to fiber-preserving symplectomorphism, noting that counterexamples exist where B does not determine the system.