The one-velocity time-independent neutron integro-differential transport equation is converted into an integral equation by the use of a homogeneous Green's function. The neutron flux, Green's function, and source are expanded in spherical harmonics. The integrations over the angles are carried out by the use of the spherical harmonic orthogonality relation. The net result is a set of coupled integral equations in the flux angular moments. Relations that give the Green's function angular moments are derived for any nonreentrant geometry and all boundary conditions applicable to the neutron transport equation. The conditions for which the scalar flux and some of the flux higher moments can be calculated exactly are discussed. Sample problems of unit slab cells that meet these conditions, are solved. The results are found to be in excellent agreement with those of the DS16 and the TRANVAR codes. A method to estimate the effect of the flux non-zeroth angular moments and the spatial truncation errors on the scalar flux is introduced. A sample problem of a heterogeneous unit slab cell is presented. It is found that the errors in the scalar flux due to neglecting the flux non-zeroth angular moments and the spatial truncation error are each of the order of 0.03% for this problem.