  This paper deals with a hyperbolic system of two nonlinear conservation laws,
where the phase space contains two contact manifolds. The governing equations
are modelling bidisperse suspensions, which consist of two types of small
particles that are dispersed in a viscous fluid and differ in size and
viscosity. For certain parameter choices quasi-umbilic points and a contact
manifold in the interior of the phase space are detected. The dependance of the
solutions structure on this contact manifold is examined. The elementary waves
that start in the origin of the phase space are classified. Prototypic Riemann
problems that connect the origin to any point in the state space and that
connect any state in the state space to the maximum line are solved
semi-analytically.
