Your system of PDE's appears to be of real principal type, as defined by Hormander. Hormander studied the singularities of distributional solutions to such a PDE and how they propagate. This in turn leads to an a regularity theorem for a compactly supported distributional solution on a bounded open domain. Since the adjoint of a PDO of real principal type is also of real principal type, the regularity theorem in turn implies a local existence theorem. If you google "hormander propagation of singularities" you should be able to find references. Books that discuss this include ones written in the 80's or earlier by Hormander, Treves, Chazarain-Piriou, and Michael Taylor.

Beyond that, as far as I know, very little is known about such PDE's. You might be able to find a way to use ad hoc techniques adapted to the specific PDE to do better than these rather general results.

ADDED: The determinant of the first order term of the PDO defines a function on the cotangent bundle of $R^2$, which is a symplectic manifold. The zero set of this function is called the characteristic variety. The function also defines a Hamiltonian vector field $H$ on the cotangent bundle. Any integral curve of the Hamiltonian vector field that lies in the characteristic variety is called a null bicharacterstic. Hormander defines an operator to be of real principal type relative to a domain $\Omega$, if there are no null bicharacteristics trapped over the domain.

It is easy to see that given a point where the function defined above has nonvanishing gradient at any point in the characteristic variety over that point is of real principal type with respect to any sufficiently small open neighborhood of that point. Your PDO appears to me to satisfy this.

Hormander's propagation of singularities theorem says that the wavefront set (which lies in the cotangent bundle and is a refinement of the singular set of a distribution) of a solution lies in the characteristic variety and is invariant under the flow of the Hamiltonian vector field.

So this indicates what you could try to do: Find the Hamiltonian vector field associated with your system of PDE's and study its flow. This would act as a guide to figuring out what the "right" domain for solving your PDE is. You would then try to prove an a priori regularity theorem, maybe through energy integral estimates, directly from your PDE, rather than using the full machinery of microlocal analysis and Fourier integral operators. Actually, you should probably also study the full matrix symbol of the PDE. I'm less familiar with the details of what to do here. You might want to consult early work of Nils Dencker on systems of real principal type.