  This note concerns stationary solutions of the Euler equations for an ideal
fluid on a closed 3-manifold. We prove that if the velocity field of such a
solution has no zeroes and real analytic Bernoulli function, then it can be
rescaled to the Reeb vector field of a stable Hamiltonian structure. In
particular, such a vector field has a periodic orbit unless the 3-manifold is a
torus bundle over the circle. We provide a counterexample showing that the
correspondence breaks down without the real analyticity hypothesis.
