  Understanding particle drifts in a non-symmetric magnetic field is of primary
interest in designing optimized stellarators to minimize the neoclassical
radial loss of particles. Quasisymmetry and omnigeneity, two distinct
properties proposed to ensure radial localization of collisionless trapped
particles in stellarators, have been explored almost exclusively for magnetic
fields that generate nested flux surfaces. In this work, we extend these
concepts to the case where all the field lines are closed. We then study
charged particle dynamics in the exact non-symmetric vacuum magnetic field with
closed field lines, obtained recently by Weitzner and Sengupta
(arXiv:1909.01890), which possesses X-points. The magnetic field can be used to
construct magnetohydrodynamic equilibrium in the limit of vanishing plasma
pressure. Expanding in the amplitude of the non-symmetric fields, we explicitly
evaluate the omnigeneity and quasisymmetry constraints. We show that the
magnetic field is omnigeneous in the sense that the drift surfaces coincide
with the pressure surfaces. However, it is not quasisymmetric according to the
standard definitions.
