  An asymptotic expansion is performed to obtain quasi-axisymmetric magnetic
configurations that are weakly non-axisymmetric. A large space of solutions is
identified, which satisfy the condition of quasi-axisymmetry on a single
magnetic flux surface, while (non-axisymmetric) globally quasi-axisymmetric
solutions are shown to not exist, agreeing with the conclusions of previous
theoretical work. The solutions found are shown to be geometrically constrained
at low aspect ratio or high toroidal period number. Solutions satisfying the
more general condition of omnigeneity (generalized quasi-axisymmetry) are also
shown to exist, and it is found that quasi-axisymmetric deformations can be
superposed with an omnigenous solution, while preserving the property of
omnigeneity, effectively extending the space of "good" configurations. A
numerical solution of the first order quasi-axisymmetry problem is demonstrated
and compared with solutions found with a widely used MHD equilibrium solver,
independently verifying that quasi-axisymmetry is satisfied at the appropriate
order. It is thereby demonstrated that approximately quasi-axisymmetric
solutions can be directly constructed, i.e. without using numerical search
algorithms.
