  The main purpose of this paper is to show the existence of action-angle
variables for integrable Hamiltonian systems on Dirac manifolds under some
natural regularity and compactness conditions, using the torus action approach.
We show that the Liouville torus actions of general integrable dynamical
systems have the structure-preserving property with respect to any underlying
geometric structure of the system, and deduce the existence of action-angle
variables from this property. We also discover co-affine structures on
manifolds as a by-product of our study of action-angle variables. 22 pages.
