  While several results have pointed to the existence of exactly quasisymmetric
fields on a surface (Garren & Boozer 1991a,b; Plunk & Helander 2018), we have
obtained the first such solutions using a vacuum surface expansion formalism.
We obtain a single nonlinear parabolic PDE for a function $\eta$ such the field
strength satisfies $B = B(\eta)$. Closed-form solutions are obtained in
cylindrical, slab, and isodynamic geometries. Numerical solutions of the full
nonlinear equations in general axisymmetric toroidal geometry are obtained,
resulting in a class of quasi-helical local vacuum equilibria near an
axisymmetric surface. The analytic models provide additional insight into
general features of the nonlinear solutions, such as localization of the
surface perturbations on the inboard side.
