  We find new vacuum solutions of ${\cal N}=4$ super-Yang-Mills with totally
anti-symmetric cubic soft SUSY breaking terms, or equivalently solutions of the
IKKT matrix model of type $\mathbb{R}^4_\theta \times {\cal K}_N$ with flux
terms. The solutions can be understood in terms of 4- and 6- dimensional fuzzy
branes ${\cal K}_N$ in extra dimensions, describing self-intersecting
projections of compact flag manifolds of $SU(3)$. The 6-dimensional solutions
provide a 6-fold covering of the internal space near the origin, while the
4-dimensional branes have a triple self-intersections spanning all 6 internal
directions. The solutions have lower energy than the trivial vacuum, and we
prove that there are no negative modes. The massless modes are identified
explicitly. In particular there are chiral fermionic zero modes, linking the
coincident sheets with opposite flux at the origin. They have a $\mathbb{Z}_3$
family symmetry, originating from the Weyl group rotations.
