  The condition of omnigenity is investigated, and applied to the near-axis
expansion of Garren and Boozer (1991a). Due in part to the particular
analyticity requirements of the near-axis expansion, we find that, excluding
quasi-symmetric solutions, only one type of omnigenity, namely
quasi-isodynamicity, can be satisfied at first order in the distance from the
magnetic axis. Our construction provides a parameterization of the space of
such solutions, and the cylindrical reformulation and numerical method of
Landreman and Sengupta (2018); Landreman et al. (2019), enables their efficient
numerical construction.
