The traditional Falkner-Skan boundary layer equation is revisited in this paper with the aim of obtaining a further analytic solution. When a free stream has a certain form, an exact possible solution is already known for a moving permeable wall. The purpose here is to extend this case to the condition where a velocity slip at the surface of a wedge is in contact with the flow. Such a mechanism is found to enrich the physical properties of both the momentum and thermal boundary layers resulting from the Falkner-Skan equation. It is shown that above certain critical values of prescribed physical parameters, coexisting flow solutions exist. These solutions are later fed into the energy equation to derive coexisting exact analytic temperature fields largely influenced by the presence of a wall temperature jump parameter. The wall shear stress and heat transfer rate are also obtained in closed-form formulas ready to serve engineers working in the field.