Motivated by amplitude blowup in solutions to systems of conservation laws, we consider the following (related) problems: (1) Given a frame of n vector fields in Rⁿ, when does there exist a conservative system U_t+F(U)_x=0 with the property that the Jacobian DF has the given vector fields as eigenvectors? (2) With the same setup: when does the resulting system possess a (convex) entropy? In other words, we want to understand to what extent a conservative system and its entropies are determined by the eigenframe? The analysis makes use of classic integrability theorems (Frobenius, Darboux, Cartan-Kaehler), and we will provide several examples. This is joint work with Irina Kogan (North Carolina State University).