  We construct non-vanishing steady solutions to the Euler equations (for some
metric) with analytic Bernoulli function in each three-manifold where they can
exist: graph manifolds. Using the theory of integrable systems, any admissible
Morse-Bott function can be realized as the Bernoulli function of some
non-vanishing steady Euler flow. This can be interpreted as an inverse problem
to Arnold's structure theorem and yields as a corollary the topological
classification of such solutions. Finally, we prove that the topological
obstruction holds without the non-vanishing assumption: steady Euler flows with
a Morse-Bott Bernoulli function only exist on graph three-manifolds.
