We study the spectral theory of a family of sub-Laplacians, defined on products of compact quotients of the Heisenberg group, which are examples of completely integrable sub-Riemannian manifolds. We classify all Quantum Limits of these sub-Laplacians, expressing them through a disintegration of measure result. This disintegration follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear. Our results illustrate the fact that, because of the possibly high degeneracy of the spectrum , the spectral theory of general sub-Riemannian (or subelliptic) Laplacians can be very rich: the invariance properties of the Quantum Limits which we study are related to the classical dynamics of infinitely many vector fields on the cotangent bundle of the manifold. These phenomena contrast with what happens for Riemannian Laplacians, for which any Quantum Limit is simply invariant under the geodesic flow.