We consider one typical two-parameter family of quadratic systems of 2 x 2 conservation laws, and study the geometry of the behaviour of the possible solutions of the Riemann problem near an umbilic point, following the geometric approach presented by Isaacson, Marchesin, Palmeira, Plohr, in A global formalism for nonlinear waves in conservation laws, Commun. Math. Phys. (1992). The corresponding phase portraits for the rarefaction curves, shock curves and composite curves are discussed.
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