Integrability structures of the (2 + 1)-dimensional Euler equation

We construct a local variational Poisson structure (a Hamiltonian operator) for the $(2+1)$-dimensional Euler equation in vorticity form. The inverse defines a nonlocal symplectic structure for the equation. We describe the action of this operator on the infinitesimal contact symmetries in terms of differential coverings over the Euler equation. Furthermore, we construct a nonlocal recursion operator for cosymmetries. Finally, we generalize the local variational Poisson structure for the Euler equation in vorticity form on a two-dimensional Riemannian manifold.