  We classify all the phenomenologically viable fermion mass matrices coming
from a spontaneously broken abelian symmetry $ U(1)_X, $ with one and two
additional chiral fields of opposite charges $ X=\pm 1. $ We find that the
non-trivial K\"ahler metric can fill zeroes of the fermion mass matrices up to
phenomenologically interesting values. A general anomaly analysis shows that
for one additional chiral field the only way to achieve anomaly cancellation is
by use of the Green-Schwarz mechanism. For two additional fields with $X=\pm 1$
and negative charge differences in the lepton sector the anomalies can however
be directly put to zero. This case gives a unique prediction for the ratio of
the two Higgs scalars of MSSM, ${\rm tg} \beta \sim { m_t \over m_b} (sin
\theta_c)^ 2$, where $\theta_c$ is the Cabibbo angle.
