- The conditions on shell-model Hamiltonians which have eigenstates with generalized seniority 1) v = 0 and v = 2 are stated and investigated in detail. For even semi-magic nuclei the conditions for v = 0 eigenstates give rise to a simple binding-energy formula with terms linear and quadratic in nucleon number. If the v = 2 conditions are also satisfied, constant spacings independent of nucleon number are obtained between ground states and the low-lying J = 2, 4, ..., levels. This feature is clearly demonstrated by the existence of a single-particle operator which transforms the v = 0 state into one with v = 2 and which obeys a linear equation of motion when acting on the v = 0 state. The constant spacings are obtained in the general case for one state with a given J, unlike the situation in the quasi-spin scheme in which there are n-independent separations between all levels. Examples are given of cases in which these conditions are actually fulfilled and yet in which the eigenstates are not those of the quasi-spin formalism. 1972.