  We show that one can reduce the coupled system of seven field equations of
the (3+1)-dimensional gauged Skyrme model to the Heun equation (which, for
suitable choices of the parameters, can be further reduced to the
Whittaker-Hill equation) in two non-trivial topological sectors. Hence, one can
get a complete analytic description of gauged solitons in (3+1) dimensions
living within a finite volume in terms of classic results in the theory of
differential equations and Kummer's con uent functions. We consider here two
types of gauged solitons: gauged Skyrmions and gauged time-crystals (namely,
gauged solitons periodic in time, whose time-period is protected by a winding
number). The dependence of the energy of the gauged Skyrmions on the Baryon
charge can be determined explicitly. The theory of Kummer's confluent functions
leads to a quantization condition for the period of the time-crystals.
Likewise, the theory of Sturm-Liouville operators gives rise to a quantization
condition for the volume occupied by the gauged Skyrmions. The present analysis
also discloses that resurgent techniques are very well suited to deal with the
gauged Skyrme model as well. In particular, we discuss a very nice relation
between the electromagnetic perturbations of the gauged Skyrmions and the
Mathieu equation which allows to use many of the modern resurgence techniques
to determine the behavior of the spectrum of these perturbations.
