  We give new estimates for a critical elliptic system introduced by
Rivi\`ere-Struwe in \cite{riviere_struwe} (see also the work of Rupflin
\cite{rupflin} and Schikorra \cite{schikorra_frames}), which generalises PDE
solved by harmonic (and almost harmonic) maps from a Euclidean ball $B_1 \In
\R^n$ into Riemannian manifolds. Solutions take the form $$-\Dl u = \Om.\D u $$
where $\Om$ is an anti-symmetric potential with $\Om$ and $\D u$ belonging to
the Morrey space $\M^{2,n-2}$ making the PDE critical from a regularity
perspective (classical theory gives one estimates on $\D u$ in the weak-Morrey
space $\M^{(2,\infty),n-2}$, see Sections \ref{adams_decay} and \ref{Morrey}
for definitions if necessary). We use the Coulomb frame method employed in
\cite{riviere_struwe} along with the H\"older regularity already acquired in
\cite{rupflin}, coupled with an extension of a Riesz potential estimate of
Adams \cite{adams_riesz} in order to attain estimates on $\D^2 u \in \M^{s,
n-2}$ for any $s<2$. These methods apply when $n=2$ thereby re-proving the full
regularity in this case (see \cite{Sh_To}) using Coulomb gauge methods.
Moreover they lead to a self contained proof of the local regularity of
stationary harmonic maps in high dimension (see Corollary \ref{highint}).
