  Current studies of WN Toda field theory focus on correlation functions such
that the WN highest-weight representations in the fusion channels are
multiplicity-free. In this work, we study W3 Toda 4-point functions with
multiplicity in the fusion channel. The conformal blocks of these 4-point
functions involve matrix elements of a fully-degenerate primary field with a
highest-weight in the adjoint representation of sl3, and a semi-degenerate
primary field with a highest-weight in the fundamental representation of sl3.
We show that, when the fusion rules are obeyed, the matrix elements of the
fully-degenerate adjoint field, between two arbitrary descendant states, can be
computed explicitly, on equal footing with the matrix elements of the
semi-degenerate fundamental field. Using null-state conditions, we obtain a
fourth-order Fuchsian differential equation for the conformal blocks. Using
Okubo theory, we show that, due to the presence of multiplicities, this
differential equation belongs to a class of Fuchsian equations that is
different from those that have appeared so far in WN theories. We solve this
equation, compute its monodromy group, and construct the monodromy-invariant
correlation functions.
