  The affine motion of two-dimensional (2d) incompressible fluids surrounded by
vacuum can be reduced to a completely integrable and globally solvable
Hamiltonian system of ordinary differential equations for the deformation
gradient in ${\rm SL}(2,{\mathbb R})$. In the case of perfect fluids, the
motion is given by geodesic flow in ${\rm SL}(2,{\mathbb R})$ with the
Euclidean metric, while for magnetically conducting fluids (MHD), the motion is
governed by a harmonic oscillator in ${\rm SL}(2,{\mathbb R})$. A complete
classification of the dynamics is given including rigid motions, rotating
eddies with stable and unstable manifolds, and solutions with vanishing
pressure. For perfect fluids, the displacement generically becomes unbounded,
as $t\to\pm\infty$. For MHD, solutions are bounded and generically
quasi-periodic and recurrent.
