  We study the implications of the simultaneous existence of a texture zero and
a vanishing minor in the neutrino mass matrix. There are thirty six possible
texture structures of this type, twenty one of which reduce to two texture zero
cases which have, already, been extensively studied. Of the remaining fifteen
textures only six are allowed by the current data. We examine the
phenomenological implications of the allowed texture structures for Majorana
type CP-violating phases, 1-3 mixing angle and Dirac type CP-violating phase.
All these possible textures can be generated through the seesaw mechanism and
realized in the framework of discrete abelian flavor symmetry. We present the
symmetry realization of these texture structures.
