  We prove that the correspondence between Reeb and Beltrami vector fields can
be made equivariant whenever additional symmetries of the underlying geometric
structures are considered. As a corollary of this correspondence, we show that
energy levels above the maximum of the potential energy of mechanical
Hamiltonian systems can be viewed as stationary fluid flows, though the metric
is not prescribed. In particular, we showcase the emblematic example of the
$n$-body problem and focus on the Kepler problem. We explicitly construct a
compatible Riemannian metric that makes the Kepler problem of celestial
mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold,
the Kepler-Euler flow.
