  We establish an optimal $L^p$-regularity theory for solutions to fourth order
elliptic systems with antisymmetric potentials in all supercritical dimensions
$n\ge 5$:
  $$
  \Delta^2 u=\Delta(D\cdot\nabla u)+div(E\cdot\nabla
u)+(\Delta\Omega+G)\cdot\nabla u +f
  \qquad \ {\rm{in}}\ B^n,
  $$ where $\Omega\in W^{1,2}(B^n, so_m)$ is antisymmetric and $f\in L^p(B^n)$,
and $D, E, \Omega, G$ satisfy the growth condition (GC-4), under the smallness
condition of a critical scale invariant norm of $\nabla u$ and $\nabla^2 u$.
This system was brought into lights from the study of regularity of
(stationary) biharmonic maps between manifolds by Lamm-Rivi\`ere, Struwe, and
Wang. In particular, our results improve Struwe's H\"older regularity theorem
to any H\"older exponent $\alpha\in (0,1)$ when $f\equiv 0$, and have
applications to both approximate biharmonic maps and heat flow of biharmonic
maps. As a by-product of the techniques, we also extend the $L^p$-regularity
theory of harmonic maps by Moser to Rivi\`ere-Struwe's second order elliptic
systems with antisymmetric potentials under the growth condition (GC-2) in all
dimensions, which confirms an expectation by Sharp.
