  Building upon the methods used recently, we establish the inexistence of
self-gravitating solitonic solutions for both static and strictly stationary
asymptotically flat spacetimes in generalised axion electrodynamics. This is an
Einstein-Maxwell-axion model, where the axion field $\theta$ is non-minimally
coupled to the electromagnetic field. Considering the standard QCD axion
coupling, we first present an argument for the absence of static axionic
solitons, $i.e.$ localised energy axionic-electromagnetic configurations,
yielding an everywhere regular, horizonless, asymptotically flat, static
spacetime. Then, for generic couplings $f(\theta)$ and $g(\theta)$ (subject to
mild assumptions) between the axion field and the electromagnetic field
invariants, we show there are still no solitonic solutions, even when dropping
the staticity assumption and merely requiring a strictly stationary spacetime,
regardless of the spatial isometries.
