  We find a class of exact solutions to the Lighthill Whitham Richards Payne
(LWRP) traffic flow equations. Using two consecutive lagrangian
transformations, a linearization is achieved. Next, depending on the initial
density, we either apply (again two) Lambert functions and obtain exact
formulas for the dependence of the car density and velocity on x and t, or
else, failing that, the same result in a parametric representation. The
calculation always involves two possible factorizations of a consistency
condition. Both must be considered. In physical terms, the lineup usually
separates into two offshoots at different velocities. Each velocity soon
becomes uniform. This outcome in many ways resembles the two soliton solution
to the Korteweg-de Vries equation. We check general conservation requirements.
Although traffic flow research has developed tremendously since LWRP, this
calculation, being exact, may open the door to solving similar problems, such
as gas dynamics or water flow in rivers. With this possibility in mind, we
outline the procedure in some detail at the end.
