We consider the problem of constructing systems of hyperbolic
conservation laws in one space dimension with prescribed geometry in state
space: the eigenvectors of the Jacobian of the
flux are given. This is formulated as a system of algebraic-differential
equations whose solution space is analyzed using Darboux and
Cartan-K\"ahler theorems. It turns out that already the case with three
equations is fairly complex. We give a complete list of possible scenarios
for the general systems of two and three equations and for rich systems
(i.e. when the given eigenvector fields are pairwise in involution) of
arbitrary size. As an application we characterize conservative systems
with the same eigencurves as compressible gas dynamics.
This is joint work with Kris Jenssen (Penn State University) [video]