= Solution
Parametrize the data by $(\xi,\eta,\zeta)=(\cos\theta,\sin\theta,-1)$. The determinant testing <Noncharacteristic Cauchy data> is
$$
\det\begin{pmatrix}-\sin\theta&\cos\theta\sin\theta\\
\cos\theta&\sin^2\theta\end{pmatrix}=-\sin\theta.
$$
Thus the open upper and lower semicircles are noncharacteristic, but the entire circle is not a valid noncharacteristic data curve: the determinant vanishes at $(1,0)$ and $(-1,0)$.
There is a stronger obstruction at those two points. On $y=0$, the differential equation forces $0=-z^4/3$, and hence $z=0$. This contradicts the prescribed value $z=-1$ at both endpoints. Therefore
$$
\boxed{\text{No classical solution satisfies the data on the entire circle.}}
$$
The obstruction is already present if the equation is imposed next to the circle and the solution and its first <partial derivatives> extend continuously to it.
Away from the endpoints, the <characteristic projections> are radial half-lines:
$$
(x,y)=r(\cos\theta,\sin\theta),\qquad r=\frac1{1-s\sin\theta}>0.
$$
For $\sin\theta>0$, $r$ increases with $s$; for $\sin\theta<0$, $r$ decreases with $s$. Each projection meets its own open semicircle exactly once and never crosses $y=0$. The endpoint projections themselves are single stationary points, although their lifted <characteristic curves> have varying height.
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