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