  In this article we give a universal model for geometric quantization
associated to a real polarization given by an integrable system with
non-degenerate singularities. This universal model is inspired by the cotangent
bundle, as any neighbourhood of a regular compact orbit of an integrable system
can be symplectically described as its cotangent bundle, $T^*(\mathbb T^n)$. As
any integrable system on a compact manifold will certainly have singularities
we need to make the recipe work in some singular situations. Our recipe works
for real polarizations given by integrable systems with non-degenerate
singularities. These singularities are generic in the sense that are given by
Morse-type functions and include elliptic, hyperbolic and focus-focus
singularities. Examples of systems admitting such singularities are toric,
semitoric and almost toric manifolds, as well as physical systems such as the
coupling of harmonic oscillators, the spherical pendulum or the reduction of
the Euler's equations of the rigid body on $T^*(SO(3))$ to a sphere. Our
geometric quantization formulation coincides with previous models away from the
singularities and corrects the former models for hyperbolic and focus-focus
singularities cancelling out the infinite dimensional contributions obtained by
former approaches. The geometric quantization models provided here match the
classical physical methods for mechanical systems, for instance the spherical
pendulum.
