According to the duality theory of traffic flow any well-posed kinematic wave (KW) and/or variational theory (VT) problem can be solved with the same methods either on the time-space plane or the time vs vehicle number plane. To achieve this symmetry, the model parameters and the boundary data need to be expressed in a form appropriate for each plane. It turns out, however, that when boundary data that are bounded in one plane are transformed for the other, singular points with infinite density (jumps in vehicle number) sometimes arise. These singularities require a new form of weak solution to the PDE's that we call an extended solution. Duality theory indicates that these e-solutions must exist and be unique. The paper characterizes these solutions. It shows that their only added feature is a new type of shock that can contain mass and we call a supershock. Nothing else is required. The evolution laws of these shocks are described. An exact solution method for e-problems with piecewise linear fundamental diagrams (FDs), not necessarily concave, is given. The paper also addresses the special case where the FD is concave so that VT applies. It is shown that if the FD is piecewise linear then sufficient networks used to solve VT problems with the least cost path method continue to be sufficient in the extended case. Thus, the same solution procedure produces exact results in both the conventional and extended cases.