We discuss the problem of the mode classification of adiabatic oscillations of spherically symmetric stars, with fully taking account of the perturbation to the gravitational potential. The problem is much harder than the case where we adopt the Cowling approximation (neglect of the perturbation to the gravitational potential), because we have to treat the fourth order system of ordinary differential equations, rather than the second order system. Particularly, we need to consider carefully the possibility of degeneracy in which two different eigenmodes with the same spherical degree have the same eigenfrequency. In a previous paper (Takata 2011), a scheme of the mode classification is proposed by assuming that there is no degeneracy. We make a step forward in this presentation by examining possible influences of the degeneracy on the scheme of the mode classification. We argue that the degeneracy, even if it exists, does not have any destructive effects on the mode classification. This opens the way to the mathematically complete scheme that allocates a continuous integral index to each eigenmode to discriminate among p modes, g modes, and f modes.