The elementary solutions of the two-group neutron-transport equation are used to solve critical problems for finite slabs and spheres. The half-range orthogonality properties of the basic eigenvectors are used, along with the fundamental H -matrix, to reduce the encountered system of singular integral equations to a system of Fredholm-type equations, and these final equations are solved iteratively to yield accurate predictions of the two-group values of the extrapolated endpoint and critical dimensions for a selected set of slabs and spheres.