The first line of attack on any pde is separation of variables. This system is unusual in that it is nonlinear, and yet yields separable solutions. Dym's equation is the only other nonlinear pde I know of that does that. If I were you, I would carry out a separation of variables procedure on this system and see what you get.
Question: what is the characteristic space of a system of pde's?
I'm not sure what the characteristic space is for a system of PDE's. All I know is that if it is indeed one dimensional, then there exist a solution (which may not be unique). Thise course is a killer =s
You might try looking up the term in your textbook's index (or, if you don't have a required text, ask your professor where to find a definition of that term).
As for the solution, like I said, go for a separable solution. That is, assume that
Plug this into your system, and you should be able to get all the x's to one side, and the y's to the other, in both original equations. From that, you can deduce that each of the sides is constant. Use that to convert into 4 ODE's.