  We show how a symmetry reduction of the equations for incompressible
hydrodynamics in three dimensions leads naturally to a Monge-Amp\`ere
structure, and Burgers'-type vortices are a canonical class of solutions
associated with this structure. The mapping of such solutions, which are
characterised by a linear dependence of the third component of the velocity on
the coordinate defining the axis of rotation, to solutions of the
incompressible equations in two dimensions is also shown to be an example of a
symmetry reduction The Monge-Amp\`ere structure for incompressible flow in two
dimensions is shown to be hypersymplectic.
