  It is well known that the incompressible Euler equations can be formulated in
a very geometric language. The geometric structures provide very valuable
insights into the properties of the solutions. Analogies with the
finite-dimensional model of geodesics on a Lie group with left-invariant metric
can be very instructive, but it is often difficult to prove analogues of
finite-dimensional results in the infinite-dimensional setting of Euler's
equations. In this paper we establish a result in this direction in the simple
case of steady-state solutions in two dimensions, under some non-degeneracy
assumptions. In particular, we establish, in a non-degenerate situation, a
local one-to-one correspondence between steady-states and co-adjoint orbits.
