Solution
= Solution
For the <Laplace operator>, an irrelevant nonzero factor gives the symbol $P(\xi,\eta)=\xi^2+\eta^2$. The roots in complex $\eta$ are $\eta=\pm i|\xi|$, so no fixed horizontal line avoids them for every $\xi$. A two-step staircase is
$$
\boxed{
\operatorname{Im}\eta=
\begin{cases}
2,&|\xi|\leq1,\\
0,&|\xi|>1.
\end{cases}}
$$
On the first step the roots have imaginary part in $[-1,1]$, and on the second they are nonreal, so neither step meets the zero set.