  Symmetries and conservation laws associated with the ideal Einstein-Euler
system, for stationary and axisymmetric stars, can be utilized to define a set
of flow constants. These quantities are conserved along flow lines in the sense
that their gradients are orthogonal to the four-velocity. They are also
conserved along surfaces of constant magnetic flux, making them powerful tools
to identify general features of neutron star equilibria. One important
corollary of their existence is that mixed poloidal-toroidal fields are
inconsistent with the absence of meridional flows except in some singular
sense, a surprising but powerful result first proven by Bekenstein and Oron. In
this work, we revisit the flow constant formalism to rederive this result
together with several new ones concerning both nonlinear and perturbative
magnetic equilibria. Our investigation is supplemented by some numerical
solutions for multipolar magnetic fields on top of a Tolman-VII background,
where strict power-counting of the flow constants is used to ensure a
self-consistent treatment.
