Neutrino Mass Matrices with Two Equalities Between the Elements or Cofactors

We study the implications of the existence of two equalities between the elements or cofactors of the neutrino mass matrix. There are sixty five structures of this type for each case. Phenomenological implications for unknown parameters like the effective Majorana mass of the electron neutrino and CP-violating phases are examined for the viable cases. To illustrate how such forms of the neutrino mass matrices may be realized, we also present a simple $A_4$ model for one of the classes in each case.