We discuss the final stages of double ionization of atoms in a strong linea
rly polarized laser field within a classical model. We propose that all tra
jectories leading to nonsequential double ionization pass close to a saddle
in the phase space that we identify and characterize. The saddle lies in a
two degree of freedom subspace of symmetrically escaping electrons. The di
stribution of a longitudinal moments of ions as calculated within the subsp
ace shows the double hump structure observed in experiments. Including a sy
mmetric bending mode of the electrons allows us to reproduce the transverse
ion momenta. We discuss also a path to sequential ionization and show that
it does not lead to the observed momentum distributions.