Cotangent models for integrable systems

We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on a special class called $b$-Poisson/$b$-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle coordinate theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a $b$-Poisson manifold. For this latter class of Poisson structures we define a twisted cotangent model. The equivalence with this twisted cotangent model is given by an action-angle theorem recently proved in [KMS16]. This viewpoint of cotangent models provides a new machinery to construct examples of integrable systems, which are especially valuable in the $b$-symplectic case where not many sources of examples are known. At the end of the paper we introduce non-degenerate singularities as lifted cotangent models on $b$-symplectic manifolds and discuss some generalizations of these models to general Poisson manifolds.