= Solution
The first-order <Cauchy-Kovalevskaya theorem> says that an analytic PDE system solved for the <derivative> normal to an analytic noncharacteristic initial hypersurface, with analytic initial data, has a unique local analytic solution near each point of that hypersurface. The general theorem likewise applies to higher-order equations solved for the highest normal <derivative> with analytic data for the lower normal <derivatives>.
Here the characteristic vector is $(x,y,a)$ and the normal to $z=0$ is $(0,0,1)$. Its normal component is $a$, so the theorem applies precisely when \b[$a\ne0$]. The <method of characteristics> gives $\dot x=x$, $\dot y=y$, $\dot z=a$, $\dot u=u$. Starting from $(\xi,\eta,0)$ at parameter $s=0$ gives $x=\xi e^s$, $y=\eta e^s$, $z=as$, $u=e^sf(\xi,\eta)$. Eliminating $s=z/a$ yields
$$
\boxed{u(x,y,z)=e^{z/a}f(xe^{-z/a},ye^{-z/a})}.
$$
This is locally defined wherever the arguments remain in the analytic domain of $f$, and verifies both PDE and initial data.
As $a\to0$, the normal component vanishes and the characteristic curves become tangent to the initial plane. The crossing parameter $s=z/a$ and slopes $dx/dz=x/a$, $dy/dz=y/a$ become singular. At $a=0$, arbitrary analytic data are not admissible: already on the plane they must obey $xf_x+yf_y=f$. Even compatible data do not determine <derivatives> normal to the plane. This is failure of the noncharacteristic condition, not a failure of the analytic theorem.
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