  A necessary and sufficient set of conditions for a quasisymmetric magnetic
field in the form of constraint equations is derived from first principles.
Without any assumption regarding the magnetohydrodynamic (MHD) equilibrium of
the plasma, conditions for quasisymmetry are constructed starting from the
single-particle Lagrangian to leading order. The conditions presented in the
paper are less restrictive than the set recently obtained by Burby et al.
[arXiv:1912.06468], and could facilitate ongoing efforts towards investigating
the existence of global quasisymmetric MHD equilibria. It is also shown that
quasisymmetry implies the existence of flux surfaces, regardless of whether the
field corresponds to an MHD equilibrium.
