Previously reported elementary solutions of the linear transport equation for an exponential scattering ratio c(z) = coe−z/sare shown to be complete on the range [−s/(s+1),1] provided coand s satisfy a certain inequality; thus these eigenfunctions areincompleteon the full range [−1,1]. A new set of elementary solutions with a larger spectrum is constructed and completeness is proved for the larger interval [−l,l]–but from which two certain subintervals of arbitrarily small width have been deleted, again provided coand s satisfy a certain inequality. Other half-range completeness questions are also addressed.