  In the context of minimal seesaw framework, we study the implications of
Dirac and Majorana mass matrices in which two rigid properties coexist, namely,
equalities among mass matrix elements and texture zeros. In the first part of
the study, we discuss general possibilities of the Dirac and Majorana mass
matrices for neutrinos with such hybrid structures. We then classify the mass
matrices into realistic textures which are compatible with global neutrino
oscillation data and unrealistic ones which do not comply with the data. Among
the large number of general possibilities, we find that only 6 patterns are
consistent with the observations at the level of the most minimal number of
free parameters. These solutions have only 2 adjustable parameters, so that all
the mixing angles can be described in terms of the two mass differences or pure
numbers. We analyze these textures in detail and discuss their impacts for
future neutrino experiments and for leptogenesis.
