  We show that germs of local real-analytic CR automorphisms of a real-analytic
hypersurface $M$ in $\C^2$ at a point $p\in M$ are uniquely determined by their
jets of some finite order at $p$ if and only if $M$ is not Levi-flat near $p$.
This seems to be the first necessary and sufficient result on finite jet
determination and the first result of this kind in the infinite type case.
  If $M$ is of finite type at $p$, we prove a stronger assertion: the local
real-analytic CR automorphisms of $M$ fixing $p$ are analytically parametrized
(and hence uniquely determined) by their 2-jets at $p$. This result is optimal
since the automorphisms of the unit sphere are not determined by their 1-jets
at a point of the sphere.
