The development of the study of degeneracy manifolds which contain symmetry manifolds as submanifolds in molecular systems of identical nuclei is reviewed,particularly with respect to the symmetry of electronic wave functions under permutations of identical nuclei.Early progress has been mainly in the systems X3 and X4,for which there are two simplifying features,viz: The number of internuclear distances is the same as the number of internal coordinates,and the deviations from degeneracy can often be described in terms of a two-dimensional irreducible representation of the relevant permutation group.We consider the study of systems of more than four identical nuclei,concentrating on the X5 system.We show that for such system there exists no internal coordinate system that treats all identical nuclei equivalently.The problem of constructing electronic wave functions which are single-valued with respect to the nuclear coordinates is discussed,both for the integer and half-odd integer electronic spin cases.