In this study, the applicability of ultraspherical-polynomials (P-N((lambda))) approximation as a spectral approach to spherical media transport/transfer problems is investigated for strongly anisotropic scattering media in which the scattering kernel is an admixture of linearly anisotropic and strongly backward/forward anisotropic parts. It is shown that the P-N method (corresponding to the sub-case lambda -> 1/2 in the P-N((lambda)) approximation), among all ultraspherical-polynomials methods, is the unique method converting the one-speed neutron transport equation into an algebraic eigenvalue problem in spectral functions G(n)(v). It is also demonstrated that the T-N method (corresponding to the sub-case lambda -> 0 in the P-N((lambda)) approximation) is another algebraic spectral approach suitable for spherical transport/transfer problems with restriction of strongly peaking anisotropy in scattering. The P-N((lambda)) approximation as lambda -> - 1/2 is found to be a prospective candidate, which can be converted into an algebraic spectral case by forcing some spectral functions into pre-selected proper forms. Even though the P-N((lambda)) approximations applicable as a spectral approach to the spherical media transfer problems are shown to be limited N to the sub-cases aforementioned, the unifying P-N((lambda)) formalism makes it possible to study spherical media transfer problems in general. As an example, the P-N((lambda)) method with any variable A is demonstrated to be applicable to the bare homogeneous sphere criticality problems by using pseudo-slab approach. Critical radii calculated are in very close agreement with literature values and equiconvergent.(c) 2007 Elsevier Ltd. All rights reserved.