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The mass eigenstates at extrema of the general Higgs potentials involving just the adjoint ( A ) or just the symmetric second-rank tensor ( S ) are calculated. The associated eigenvalues characterize the extrema as minima, maxima, or inflection points. A coupling between A and S is added and the changes in the eigenstates and eigenvalues are noted. Absolute minima with isotropy groups SO( N -p) X SO(p -2 X SO(2) and U(( N -p)/2) X U(p /2) are exhibited. These isotropy groups are shown to be stable under small perturbations of the potential.

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