We investigate the Majorana neutrino mass matrix $M_{ u}$ 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 $(M_{ u})_{11,12}
=0$ and $(M_{ u})_{11,13}=0$ are consistent with the current experimental constraints, and show the correlations between non-zero elements of $M_{ u}$. The ratios of non-zero elements of $M_{ u}$ are constrain in small regions, and we find simple examples of $M_{ u}$ 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 $A_5$ .
