I learned the following result from Sogge's book:
Let $P$ be a first order self-adjoint elliptic operator on $M$. Let $Q\in \Psi^{m}(M)$ be a classical $\Psi DO$ of order $m$. Then there exists a one parameter family of $\Psi DO$s: $t\rightarrow E(t)\in \Psi^{m}(M)$, depending smoothly on $t\in \mathbb{R}$, satisfying $$ [\partial_{t}-iP, E(t)]=0,E(0)=Q $$ and having for each $t\in \mathbb{R}$ the principal symbol $E_{0}(t,x,\xi)=q_{0}(\Psi_{t}(x,\xi))$ with $q_0(x,\xi)$ the principal symbol of $Q$ and $\Psi_{t}$ the Hamiltonian flow associated with $P$.
I do not know if there are more refined results directly concerning the linear equation $Qu=0$, though.