  We perform a systematic investigation of the texture structures of Majorana
neutrino mass matrix $M_{\nu}$ having two texture zeros and an equality between
two nonzero matrix elements, in the light of recent neutrino oscillation data.
Among forty-two possible textures, it is found that only eight textures are
compatible with the current experimental data at 3$\sigma$ C.L. Out of these
phenomenologically viable textures, six follow normal mass ordering while
remaining two satisfy the inverted mass ordering of neutrino mass spectrum. In
the numerical analysis, we carry out a scan over the possible space of all
viable patterns. We present the implications of each allowed patterns for three
mixing angles (solar, reactor and atmospheric), leptonic CP-violation, neutrino
mass scale and the neutrinoless double beta decay indicating strong
correlations between oscillation parameters. The symmetry realization of one of
the viable textures is also presented.
