Conical intersections play an essential role in electronically nonadiabatic processes. For molecules with an odd number of electrons the spin-orbit interaction produces essential changes in the topography and connectivity of points of conical intersection. In the nonrelativistic case, or when the molecule has an even number of electrons, eta, the dimension of the branching space, the space in which the conical topography is evinced, is 2. By contrast, for molecules with an odd number of electrons the branching space is 5 dimensional (eta = 5) in general, or 3-dimensional (eta = 3) when C-s symmetry is present. Recently, we have used degenerate perturbation theory to obtain analytic representations of the energy, and of the derivative couplings, and a "rigorous" diabatic basis in the vicinity of a conical intersection for the eta = 3 case. Here, we extend this analysis to the general, no symmetry, eta = 5, case. The perturbative results provide valuable insights into the nature of this singular point.