We study vacuum states and symmetric fermions in the equivariant dimensional reduction of the Yang-Mills-Dirac theory over the six-dimensional homogeneous space SU(3)/U(1)×U(1) endowed with a family of SU(3) structures including a nearly Kähler structure. We derive the fixed tree-level scalar potentials of the induced Yang-Mills-Higgs theory and compute the dynamically generated gauge and Higgs boson masses as functions of the metric moduli of the coset space. We find an integrable subsector of the Higgs field theory that is governed by a sine-Gordon–type model whose topological soliton solutions are determined nonperturbatively by the gauge coupling and that tunnel between families of infinitely degenerate vacua. The reduction of the Dirac action for symmetric fermions yields exactly massless chiral fermions containing subsectors that have fixed tree-level Yukawa interactions. We compute dynamical fermion mass matrices explicitly and compare them at different points of the moduli space, some of which support consistent heterotic flux vacua.