steady and unsteady Euler eq problem

AO

the unsteady euler eq is hyperbolic and Riemman problem solver is widely used in solving the hyperbolic equation by FVM. But what's class of steady euler equation? I am working on solving shallow water eq, which I think the classification is dependent of frude number, maybe hyperbolic, maybe elliptical or parobolic. So I personally think for steady euler eq or shallow water eq is unsolvable, some body can give some reliable paper or reference to prove what i thought is wrong or not. thanks .