Several new aspects of the FNmethod are reported, and the method is used to solve accurately some benchmark problems in radiative transfer that have been posed in the literature. More specifically, modifications in the method are made so that cases of pure scattering can be solved, and an improved way to compute certain required integrals is given. In addition, a two-term recursion formula that provides a computationally attractive alternative to previously used three-term recursion formulas for generating matrix elements basic to the method is established. Also, a recently developed improvement in the way the method is used to compute the radiation intensity is generalized and thus made available for all component problems in a Fourier decomposition of the complete radiation field. The numerical results given are focused on a collection of test problems including five (for a Haze L phase function and a Cloud C1phase function) posed by the Radiation Commission of the International Association of Meteorology and Atmospheric Physics.