  We investigate the prospects for scenarios with vanishing determinant
neutrino mass matrices and vanishing $\theta_{13}$ mixing angle. Normal and
inverse mass hierarchies are considered separately. For normal hierarchy it is
found that neutrinoless double beta decay cannot be observed by any of the
present or next generation experiments. For inverse hierarchy the neutrinoless
double beta decay is, on the contrary, accessible to experiments. We also
analyse for both hierarchies the case for texture zeros and equalities between
mass matrix elements. No texture zeros are found to be possible nor any such
equalities, apart from the obvious ones.
