  We present a detailed solution of the active interface equations in the
inviscid limit. The active interface equations were previously introduced as a
toy model of membrane-protein systems: they describe a stochastic interface
where growth is stimulated by inclusions which themselves move on the
interface. In the inviscid limit, the equations reduce to a pair of coupled
conservation laws. After discussing how the inviscid limit is obtained, we turn
to the corresponding Riemann problem: the solution of the set of conservation
laws with discontinuous initial condition. In particular, by considering two
physically meaningful initial conditions, a giant trough and a giant peak in
the interface, we elucidate the generation of shock waves and rarefaction fans
in the system. Then, by combining several Riemann problems, we construct an
oscillating solution of the active interface with periodic boundaries
conditions. The existence of this oscillating state reflects the reciprocal
coupling between the two conserved quantities in our system.
