For kernels appearing in the system of integral equations for Legendre moments of the angular flux, we propose a factorized form that also accounts for the anisotropy of scattering and works in the original Euclidean space. The stationary problem in the above simplified mathematical formulation for monoenergetic neutrons is then solved by a DKPL technique, i.e., a suitable basis is defined, in terms of Legendre polynomials of the space variables, and the corresponding Fourier series development is adopted for the space distribution to reduce the system of integral equations for such unknowns to an algebraic system on the unknown coefficients of their Fourier series expansion inside the homogeneous parallelepiped. This expansion converges in the mean and point-wise uniformly to the exact solution. Both critical and subcritical physical situations are considered, and accurate numerical results for isotropic scattering are obtained.