  We characterize, using commuting zero-flux homologies, those
volume-preserving vector fields on a $3$-manifold that are steady solutions of
the Euler equations for some Riemannian metric. This result extends Sullivan's
homological characterization of geodesible flows in the volume-preserving case.
As an application, we show that the steady Euler flows cannot be constructed
using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in
higher dimensions are also proved.
