Riemann problem solutions for a balance law under Dirac-Delta source with a discontinuous flux

We study the Riemann problem for a new model on immiscible vertical two-phase flow under point injection. The point injection is modeled by a Dirac delta-source term as well as by a spatially discontinuous flux function, which defines two fluxes, one on each side of the discontinuity. The solutions comprise up to three wave groups: downward waves, a stationary shock and upward waves. Owning the interplay between the Dirac delta-source and the discontinuous flux, there is no standard entropy condition for the stationary shock (flux's connections). An entropy condition was deduced based on impinging characteristics and perturbation of the constant solution. This condition leads to shocks that do not satisfy the classical Lax's conditions - even for arbitrarily small shocks - and may also have no viscous profile. The Rankine-Hugoniot condition - embedding the Dirac delta-source - and the entropy condition are geometrically represented by "Flux Projections" that support the analytical method proposed in this paper. We then obtain a & Laplacetrf;(1)(loc) analytic solution for all initial value problems. We verify the entropy condition using a Lagrangian-Eulerian scheme recently introduced in the literature, which is based in the new concept of no-flow curves. Analytic and numerical solutions fit.