  A generic valon potential invariant under U(1)\x SO(3) is used to determine
whether the horizontal symmetry of leptons is $A_4$, $S_4$, or some other
finite subgroups of SO(3). Valons in the potential are assigned SO(3) spins 0,
1, and 2, for these are the only ones that can couple to the three generation
of fermions with horizontal spin 1. This potential causes a breakdown into
three possible phases with three different symmetries. Phase I has an $A_4$
symmetry suitable for leptonic mixing. Phase II has an SO(2) symmetry and phase
III has a $Z_2\x Z_2$ symmetry, both capable of describing Cabibbo mixing of
quarks. Phase II has to be rejected on phenomenological ground, but phase III
yields block diagonal and hierarchical mass matrices appropriate for quarks. No
other non-abelian symmetry such as $S_4$ is present.
