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

$SO(2)$ gauged Skyrmions in $4+1$ dimensions