Approximate symmetries of guiding-centre motion

Quasisymmetry builds a third invariant for charged-particle motion besides energy and magnetic moment. We address quasisymmetry at the level of approximate symmetries first-order guiding-centre motion. find that conditions to leading order are same as exact if one insists symmetry is purely spatial. also generalise allow phase-space symmetries, derive weaker conditions. The latter recover "weak quasisymmetry" subcase, thus we prove it spatial only order, but implies existence wider class independent conserved quantities. Finally, demonstrate magnetohydrostatics imposes order.