  We study boundary regularity of maps from two-dimensional domains into
manifolds which are critical with respect to a generic conformally invariant
variational functional and which, at the boundary, enter perpendicularly into a
support manifold. For example, harmonic maps, or $H$-surfaces, with a partially
free boundary condition.
  In the interior it is known, by the celebrated work of Riviere, that these
maps satisfy a system with an antisymmetric potential, from which one can
derive regularity of the solution. We show that these maps satisfy along the
boundary a system with a nonlocal antisymmetric boundary potential which
contains information from the interior potential and the geometric Neumann
boundary condition. We then proceed to show boundary regularity for solutions
to such systems.
