  Let M be a connected real-analytic hypersurface in two dimensional complex
space, $\mathbb C^2$, containing a connected complex hypersurface E, and let f
be a smooth CR mapping sending M into another real-analytic hypersurface M' in
$\mathbb C^2$. In this paper, we prove that if f does not collapse E to a point
and does not collapse M into the image of E, and if the Levi form of M vanishes
to first order along E, then f is real-analytic in a neighborhood of E. In
general, the corresponding statement is false if the Levi form of M vanishes to
second order or higher, in view of an example due to the author. We also show
analogous results in higher dimensions provided that the target M' satisfies a
certain nondegeneracy condition.
  The main ingredient in the proof, which seems to be of independent interest,
is the prolongation of the system defining a CR mapping sending M into M' to a
Pfaffian system on M with singularities along E. The nature of the singularity
is described by the order of vanishing of the Levi form along E.

Complex Variables