  Stellarators are generically small current and low plasma beta ($\beta=
p/B^2\ll1$) devices. Often the construction of vacuum magnetic fields with good
magnetic surfaces is the starting point for an equilibrium calculation.
Although in cases with some continuous spatial symmetry flux functions can
always be found for vacuum magnetic fields, an analogous function does not, in
general, exist in three dimensions. This work examines several simple
equilibria and vacuum magnetic field problems with the intent of demonstrating
the possibilities and limitations in the construction of such states. Starting
with a simple vacuum magnetic field with closed field lines in a topological
torus (toroidal shell with a flat metric), we obtain a self-consistent formal
perturbation series using the amplitude of the nonsymmetric vacuum fields as a
small parameter. We show that systems possessing stellarator symmetry allow the
construction order by order. We further indicate the significance of
stellarator symmetry in the amplitude expansion of the full ideal
magnetohydrodynamics (MHD) problem as well. We then investigate the conditions
that guarantee neighboring flux surfaces given the data on one surface, by
expanding in the distance from that surface. We show that it is much more
difficult to find low shear vacuum fields with surfaces than force-free fields
or ideal MHD equilibrium. Finally, we demonstrate the existence of a class of
vacuum magnetic fields, analogous to `snakes' observed in tokamaks, which can
be expanded to all orders.
