A nonlinear Hamiltonian structure for the Euler equations

The Euler equations for inviscid incompressible fluid flow have a Hamiltonian structure in Eulerian coordinates, the Hamiltonian operator, though, depending on the vorticity. Conservation laws arise from two sources. One parameter symmetry groups, which are completely classified, yield the invariance of energy and linear and angular momenta. Degeneracies of the Hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point-wise conservation of vorticity. It is conjectured that no further conservation laws exist, indicating that the Euler equations are not completely integrable, in particular, do not have soliton-like solutions.