We consider the Cauchy problem for a strictly hyperbolic \(2\times 2\) system of conservation laws in one space dimension \( u_t+[F(u)]_x=0, u(0,x)=\bar u(x),\) which is neither linearly degenerate nor genuinely non-linear. We make the following assumption on the characteristic fields. If \(r_i(u), \ i=1,2,\) denotes the \(i\)-th right eigenvector of \(DF(u)\) and \(\lambda_i(u)\) the corresponding eigenvalue, then the set \(\{u : \nabla \lambda_i \cdot r_i (u) = 0\}\) is a smooth curve in the \(u\)-plane that is transversal to the vector field \(r_i(u)\).

Systems of conservation laws that fulfill such assumptions arise in studying elastodynamics or rigid heat conductors at low temperature.

For such systems we prove the existence of a closed domain \(\mathcal{D} \subset L^1,\) containing all functions with sufficiently small total variation, and of a uniformly Lipschitz continuous semigroup \(S:\mathcal{D} \times [0,+\infty)\rightarrow \mathcal{D}\) with the following properties. Each trajectory \(t \mapsto S_t \bar u\) of \(S\) is a weak solution of (1). Vice versa, if a piecewise Lipschitz, entropic solution \(u= u(t,x)\) of (1) exists for \(t \in [0,T],\) then it coincides with the trajectory of \(S\), i.e. \(u(t,\cdot) = S_t \bar u.\)

This result yields the uniqueness and continuous dependence of weak, entropy-admissible solutions of the Cauchy problem (1) with small initial data, for systems satisfying the above assumption.