  In this article, we explore the low energy structure of a $U(3)$ gauge theory
over spaces with fuzzy sphere(s) as extra dimensions. In particular, we
determine the equivariant parametrization of the gauge fields, which transform
either invariantly or as vectors under the combined action of $SU(2)$ rotations
of the fuzzy spheres and those $U(3)$ gauge transformations generated by $SU(2)
\subset U(3)$ carrying the spin $1$ irreducible representation of $SU(2)$. The
cases of a single fuzzy sphere $S_F^2$ and a particular direct sum of
concentric fuzzy spheres, $S_F^{2 \, Int}$, covering the monopole bundle
sectors with windings $\pm 1$ are treated in full and the low energy degrees of
freedom for the gauge fields are obtained. Employing the parametrizations of
the fields in the former case, we determine a low energy action by tracing over
the fuzzy sphere and show that the emerging model is abelian Higgs type with
$U(1) \times U(1)$ gauge symmetry and possess vortex solutions on ${\mathbb
R}^2$, which we discuss in some detail. Generalization of our formulation to
the equivariant parametrization of gauge fields in $U(n)$ theories is also
briefly addressed.
