  We construct new stationary weak solutions of the 3D Euler equation with
compact support. The solutions, which are piecewise smooth and discontinuous
across a surface, are axisymmetric with swirl. The range of solutions we find
is different from, and larger than, the family of smooth stationary solutions
recently obtained by Gavrilov and Constantin-La-Vicol; in particular, these
solutions are not localizable. A key step in the proof is the construction of
solutions to an overdetermined elliptic boundary value problem where one
prescribes both Dirichlet and (nonconstant) Neumann data.
