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