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We present the general form of the unitary matrices keeping invariant theMajorana neutrino mass matrix of specific texture suitable for explainingoscillation data. In the case of the Tri-bimaximal pattern with two degeneratemasses we give a specific realization of the underlying $U(1)$ symmetry whichcan 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 accommodatethe hierarchy of the charged lepton masses. The lepton mass spectrum can alsobe achieved in another model extending the SM with three SM-singlet scalarstransforming non trivially under the flavor symmetry. We discuss how such amodel has room for generating enough baryon asymmetry through leptogenesis inthe framework of Type-I and II seesaw mechanisms.

Form Invariance and Symmetry in the Neutrino Mass Matrix