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Quasisymmetry and omnigeneity of an equilibrium magnetic field are twodistinct properties proposed to ensure radial localization of collisionlesstrapped particles in any stellarator. These constraints are incompletelyexplored, but have stringent restrictions on a magnetic geometry. This workemploys an analytic approach to understand the implications of the constraints.The particles move in an intrinsically three dimensional equilibrium whoserepresentation is given by earlier work of Weitzner and its extension here. Fordeeply trapped particles a local equilibrium expansion around a minimum of themagnetic field strength along a magnetic line suffices. This analyticalnon-symmetric equilibrium solution, enables explicit representation of theconstraints. The results show that it is far easier to satisfy the omnigeneitycondition than the quasisymmetry requirement. Correspondingly, there exists alarge class of equilibrium close to quasisymmetry that remain omnigeneous whileallowing inclusion of error fields, which may destroy quasisymmetry.

Radial confinement of deeply trapped particles in a non-symmetric magnetohydrodynamic equlibrium