  This paper is part of a series concerning the isospectral problem for an
ellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipse
with Dirichlet or Neumann boundary conditions. Using many classical results on
ellipse eigenfunctions, we determine the microlocal defect measures of the
Cauchy data of the eigenfunctions. The ellipse has integrable billiards, i.e.
the boundary phase space is foliated by invariant curves of the billiard map.
We prove that, for any invariant curve $C$, there exists a sequence of
eigenfunctions whose Cauchy data concentrates on $C$. We use this result to
give a new proof that ellipses are infinitesimally spectrally rigid among
$C^{\infty}$ domains with the symmetries of the ellipse.
