  We study the simplest $SO(2)$ gauged $O(5)$ Skyrme models in $4+1$ (flat)
dimensions. In the gauge decoupled limit, the model supports topologically
stable solitons (Skyrmions) and after gauging, the static energy of the
solutions is bounded from below by a "baryon number". The studied model
features both Maxwell and Maxwell--Chern-Simons dynamics. The considered
configurations are subject to bi-azimuthal symmetry in the ${\mathbb R}^4$
subspace resulting in a two dimensional subsystem, as well as subject to an
enhanced symmetry relating the two planes in the ${\mathbb R}^4$ subspace,
which results in a one dimensional subsystem. Numerical solutions are
constructed in both cases. In the purely magnetic case, fully bi-azimuthal
solutions were given, while electrically charged and spinning solutions were
constructed only in the radial (enhanced symmetric) case, both in the presence
of a Chern-Simons term, and in its absence. We find that, in contrast with the
analogous models in $2+1$ dimensions, the presence of the Chern-Simons term in
the model under study here results only in quantitative effects.
