Noncommutative Integrable Systems on $b$-symplectic Manifolds

In this paper we study noncommutative integrable systems on $b$-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a $b$-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the $b$-symplectic structure.