Abstract
We study the orbit structure of a vector fieldV defined on a three-dimensional Riemannian manifold which satisfiesV ^ curlV=0. Such a vector field represents the velocity of a stationary solution of Euler’s equation for a perfect fluid. In addition to several other results, we show that if the vector field admits a first integral, then each level set is toroidal and the induced flow on the level set is either periodic or conditionally periodic.
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