  We present the general form of the unitary matrices keeping invariant the
Majorana neutrino mass matrix of specific texture suitable for explaining
oscillation data. In the case of the Tri-bimaximal pattern with two degenerate
masses we give a specific realization of the underlying $U(1)$ symmetry which
can be uplifted to a symmetry in a complete theory including charged leptons.
For this, we present a model with three light SM-like
  Higgs doublets and one heavy Higgs triplet and find that one can accommodate
the hierarchy of the charged lepton masses. The lepton mass spectrum can also
be achieved in another model extending the SM with three SM-singlet scalars
transforming non trivially under the flavor symmetry. We discuss how such a
model has room for generating enough baryon asymmetry through leptogenesis in
the framework of Type-I and II seesaw mechanisms.
