  We rereview the classical Cauchy-Kovalevskaya theorem and the related
uniqueness theorem of Holmgren, in the simple setting of powers of the
Laplacian and a smooth curve segment in the plane. As a local problem, the
Cauchy-Kovalevskaya and Holmgren theorems supply a complete answer to the
existence and uniqueness issues. Here, we consider a global uniqueness problem
of Holmgren's type. Perhaps surprisingly, we obtain a connection with the
theory of quadrature identities, which demonstrates that rather subtle
algebraic properties of the curve come into play. For instance, if $\Omega$ is
the interior domain of an ellipse, and $I$ is a proper arc of the ellipse
$\partial\Omega$, then there exists a nontrivial biharmonic function $u$ in
$\Omega$ which vanishes to degree three on $I$ (i.e., all partial derivatives
of $u$ of order $\le2$ vanish on $I$) if and only if the ellipse is a circle.
Finally, we consider a three-dimensional case, and analyze it partially using
analogues of the square of the 2X2 Cauchy-Riemann operator.
