  We will study some possible non universal corrections that could affect
single measurements of the Weinberg angle sin^2(theta) (leaving the full
average essentially unchanged). We will concentrate on the class of the models
that introduces a new light gauge boson and an extended Higgs sector, but we
restrict the Higgs charges choice in order to automatically achieve zero mixing
at the tree level between the Z_0 and the Z'. This will more naturally keep
small the universal effects with a relatively light Z', but it will imply only
vector-like coupling between this boson and the standard fermions. A SO(10)
gauge boson which distinguishes between families can have such properties and
could explain a deviation in the tau forward-backward asymmetry without
affecting the tau polarization measurements at LEP.
