In this paper, we consider the following Riemann problem: 1 \(\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial U}{\partial t}+\dfrac{\partial }{\partial x}F(U)=0,\\ U(x,0)={\left\{ \begin{array}{ll} U_L \ \text {if} \ x<0,\\ U_R \ \text {if} \ x>0, \end{array}\right. } \end{array}\right. } \end{aligned}\) where \(U(x,t)=(u(x,t),v(x,t))^\intercal \) , u(x, t), v(x, t) are unknown scalar functions defined on \({\mathbb {R}} \times [ 0, +\infty [\) with values in \({\mathbb {R}}\) , \( F(U) = (f(u), g(v))^\intercal \) , where f and g are sufficiently regular (continuous and at least twice differentiable), \( U_L \) and \( U_R \) are arbitrary constants in \({\mathbb {R}}^2\) . The system (1) is not strictly hyperbolic because its eigenvalues are real and can be equal. However, it is equivalent to two scalar hyperbolic Riemann problems. This study compares the solution of the system and its characteristics with those of the scalar cases. In particular, we establish relationships that may exist between the intermediate states and their numbers, as well as the nature of the waves (rarefaction, shock, contact discontinuity, or other waves). The comparison of the admissibility criteria of Lax and Oleinik allows for the definition of the correct admissible path in the system. Such a comparison allows us to highlight important properties.
