 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 isOn 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.