I have now strong doubts about your answer for the case $c=-1$ In fact, I believe that there is a self-adjoint extension in that case.

I looked at the Sjöstrand reference and did not see anything on the case $c\in (-\infty, 0)$. He studies in that book the case $c\in \mathbb C\backslash(-\infty, 0[$ which indeed are all non-self-adjoint if $c$ is not positive, but the case $c\in (-\infty, 0)$ is apparently not tackled in that reference. Thanks in advance for your help.

OK, I want ellipticity and the characteristic set to be a manifold, which is not the case of your example $\xi=\pm x$.

I mean that $a=a_1+a_0$ with $a_j\in \Sigma^j$ is such that $\vert \nabla a_1\vert^2$ is elliptic in $\Sigma_1$.

Thanks, I modified the question with a real principal type symbol $a$.

I have no more heuristic explanations. If you want to enter the details, I would recommend the paper MR2124585,Transport equations with partially BV velocities, by N. Lerner and the L. Ambrosio article quoted there.

Obviously you can work in a coordinate chart, assume that $c=0$ and $$ F(x)=-x_1^2-\dots -x_k^2+x_{k+1}^2+\dots+x_{n}^2=-\vert x'\vert^2+\vert x''\vert^2, $$ $$ M_{-\epsilon}=\{(x',x''), \vert x''\vert^2+\epsilon\le\vert x'\vert^2 \},\quad M_{\epsilon}=\{(x',x''), \vert x''\vert^2\le\vert x'\vert^2 +\epsilon\}. $$