  We study the problem of constructing systems of hyperbolic conservation laws
in one space dimension with prescribed eigencurves, i.e. the eigenvector fields
of the Jacobian of the flux are given. We formulate this as a typically
overdetermined system of equations for the eigenvalues-to-be. Equivalent
formulations in terms of differential and algebraic-differential equations are
considered. The resulting equations are then analyzed using appropriate
integrability theorems (Frobenius, Darboux and Cartan-Kahler). We give a
complete analysis of the possible scenarios, including examples, for systems of
three equations. As an application we characterize conservative systems with
the same eigencurves as the Euler system for 1-dimensional compressible gas
dynamics. The case of general rich systems of any size (i.e. when the given
eigenvector fields are pairwise in involution; this includes all systems of two
equations) is completely resolved and we consider various examples in this
class.
