= Solution
The order-$m$ <principal symbol> of the <formal adjoint> is $p_m^*=(-1)^mp_m$. Derivatives falling on coefficients contribute only lower-order terms. Thus the original plane remains a <non-characteristic hypersurface>.
For the curved face use the defining function $\rho_\varepsilon=x+y^2-\varepsilon$, whose conormal is $(1,2y)$. On the face with $x>0$, $|y|<\sqrt\varepsilon$ and $0<x\le\varepsilon$. Therefore $(x,y,1,2y)$ tends uniformly to $(0,0,1,0)$ as $\varepsilon\downarrow0$. Since $p_m^*(0,0;1,0)\ne0$, continuity gives
$$
\boxed{|p_m^*(x,y;1,2y)|\ge c>0\quad\text{on every sufficiently small curved face}.}
$$
The same argument supplies a common neighborhood where the transformed coefficient of the highest <normal derivative> is nonzero, permitting the <real analytic> normal form used below.
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