The combination of strongly forward and backward scattering with usual linearly anisotropic scattering has been studied for one-speed neutrons in infinite slabs with vacuum boundary conditions. For critical systems the criticality factor is given for two slab thicknesses and different degrees of the anisotropic scattering. The special case when the forward and backward scattering completely dominate over the 'ordinary' scattering is discussed in detail. The time decay of a neutron field in a moderator has also been studied. It is shown that the decay constant for thin slabs cannot exceed a certain limit determined by the degree of strongly forward and backward scattering.