In this article, we study the initial-value problem for two first-order systems in non-conservative form. The first system arises in elasto-dynamics and belongs to the class of strictly hyperbolic, genuinely nonlinear systems. The second system has repeated eigenvalues and an incomplete set of right eigenvectors. Solutions to such systems are expected to develop singular concentrations. Existence of singular solutions to both the systems have been shown using the method of weak asymptotics. The

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... second system has been shown to develop singular concentrations even from Riemann-type initial data. The first system differing from the second in having an extra term containing a positive constant k, the solution constructed for the first system have been shown to converge to the solution of the second as k tends to 0. , E 2 (u, σ) = 1 −k. 2000 Mathematics Subject Classification. 35L65, 35L67. Key words and phrases. Hyperbolic systems of conservation laws; δ-shock wave type solution; weak asymptotic method. Now letting k → 0, we see that the eigenvalues λ 1 (u, σ) and λ 2 (u, σ) tend to coincide. In particular, taking k = 0 in (1.1) we arrive at the system ∂u ∂t + u ∂u ∂x − ∂σ ∂x = 0, ∂σ ∂t + u ∂σ ∂x = 0. (1.2) which has repeated eigenvalues λ 1 (u, σ) = λ 2 (u, σ) = u and an incomplete set of