  We reconsider the earlier found solutions of the Knizhnik-Zamolodchikov (KZ)
equations describing correlators based on the admissible representations of
$A_1^{(1)}$. Exploiting a symmetry of the KZ equations we show that the
original infinite sums representing the 4-point chiral correlators can be
effectively summed up. Using these simplified expressions with proper choices
of the contours we determine the duality (braid and fusion) transformations and
show that they are consistent with the fusion rules of Awata and Yamada. The
requirement of locality leads to a 1-parameter family of monodromy (braid)
invariants. These analogs of the ``diagonal'' 2-dimensional local 4-point
functions in the minimal Virasoro theory contain in general non-diagonal terms.
They correspond to pairs of fields of identical monodromy, having one and the
same counterpart in the limit to the Virasoro minimal correlators.
