  In this paper we present some classification results for the steady Euler
equations in two-dimensional exterior domains with free boundaries. We prove
that, in an exterior domain, if a steady Euler flow devoid of interior
stagnation points adheres to slip boundary conditions and maintains a constant
norm on the boundary, along with certain additional conditions at infinity,
then the domain is the complement of a disk, and the flow is circular, namely
the streamlines are concentric circles. Additionally, we establish that in the
entire plane, if all the stagnation points of a steady Euler flow
coincidentally form a disk, then, under certain additional reasonable
conditions near the stagnation points and at infinity, the flow must be
circular. The proof is based on a refinement of the method of moving planes.
