  We investigate the texture structures of lepton mass matrices with four
(five) non-zero elements in the charged lepton mass matrix and three (four)
vanishing cofactors in the neutrino mass matrix. Using weak basis
transformations, all possible textures for three and four vanishing cofactors
in $M_{\nu}$ are grouped into 7 classes and predictions for the unknown
parameters such as the Dirac CP violating phase and the effective Majorana mass
for the phenomenologically allowed textures have been obtained. We, also,
illustrate how such texture structures can be realized using discrete Abelian
flavor symmetries.
