We consider an exact reduction of a model of Field Dislocation Mechanics to a scalar problem in one spatial dimension and  investigate the existence of static and slow, rigidly moving single or collections of planar screw dislocation walls in this  setting. Two classes of drag coefficient functions are considered,  namely those with linear growth near the origin and those with  constant or more generally sublinear growth there. A mathematical  characterisation of all possible equilibria of these screw wall  microstructures is given. We also prove the existence of travelling   wave solutions for linear drag coefficient functions at low wave  speeds and rule out the existence of nonconstant bounded travelling   wave solutions for sublinear drag coefficients functions. It turns  out that the appropriate concept of a solution in this scalar case   is that of a viscosity solution. The governing equation is not  proper and it is shown that no comparison principle holds. The   findings indicate a short-range nature of the stress field of the  individual dislocation walls, which indicates that the nonlinearity  present in the model may have a stabilising effect.