Second level semi-degenerate fields in W3 Toda theory: matrix element and differential equation

In a recent study we considered W-3 Toda 4-point functions that involve matrix elements of a primary field with the highest-weight in the adjoint representation of (sic)(3). We generalize this result by considering a semi-degenerate primary field, which has one null vector at level two. We obtain a sixth-order Fuchsian differential equation for the conformal blocks. We discuss the presence of multiplicities, the matrix elements and the fusion rules.