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We investigate the Majorana neutrino mass matrix $M_{\nu}$ with one parameter in the context of two texture zeros and its symmetry realization by non-Abelian discrete symmetry. From numerical calculation, we confirm that the textures $(M_{\nu})_{11,12}=0$ and $(M_{\nu})_{11,13}=0$ are consistent with the current experimental constraints, and show the correlations between non-zero elements of $M_{\nu}$. The ratios of non-zero elements of $M_{\nu}$ are constrain in small regions, and we find simple examples of $M_{\nu}$ with one real mass parameter. We also discuss symmetry realization of the mass matrix by the type-II seesaw mechanism based on the binary icosahedral symmetry $A_5'$ .

One-parameter Neutrino Mass Matrix and Symmetry Realization