Let $M$ be a compact, connected, oriented manifold. Let $p\in C^\infty(T^*M)$ be a real and positive principal symbol of some degree $2d\in 2\mathbb{N}$ elliptic differential operator $P\in \mathrm{Diff}^{2d}(M)$. That is, $p$ is smooth, $p(x,\alpha\cdot \xi) = \alpha^{2d} p(x,\xi)$ for all $(x,\xi) \in T^*M$ and $\alpha\in \mathbb{R}$. In local coordinates, $p$ is a homogeneous polynomial in the cotangent variables. Taking $p$ as a Hamiltonian on $T^*M$, the resulting flow $\varphi_{t} : T^*M \to T^* M$ is complete, since $p$ is homogeneous and $M$ is compact. This is the well-known bicharacteristic flow, which would roughly tell us how $P$ would propagate singularities if it were not elliptic. In the special case that $p(x,\xi)= \frac{1}{2} |\xi|_{g}^2$ in the dual of some metric $g$ on $M$ (this is the symbol of $-\frac{1}{2}\Delta_g$), the projection of this flow via $\pi_M : T^*M \to M : (x,\xi) \mapsto x$ is equivalent to the exponential map, after identifying $TM$ with $T^*M$.
My question is then, what happens if $d>1$? Can we still get injectivity of \begin{align} \{ \xi \in T^*_x M\setminus \{ 0\} \, | \, p(x,\xi) < \epsilon \} \to M : \xi \mapsto (\pi_M \circ \varphi_1)(x,\xi) \end{align} for some $\epsilon >0$ small enough? Looking at a local Taylor expansion, I suspect that this might be possible under some extra conditions, for example that $T_x^* M \to T_x M : \xi \mapsto \partial_\xi p(x,\xi)$ (the Legendre transformation) is invertible for all $x\in M$. There has to be something about this in the literature. Alas, my search has been mostly fruitless. The question is both general, but in a way also rather specific. Maybe I do not fully understand the literature, or have just not read enough. I am interested in knowing whether such flows have nice properties that resemble the geodesic flow. Assuming injectivity, does the travel time (analogous to geodesic distance) locally look like the symbol itself?