Dispersion curves of metamaterial steerable antennas composed of two-dimensional arrays of metallic unit structures with the C4v and C6v symmetries are calculated both qualitatively by the tight-binding approximation and quantitatively by the finite-difference time-domain method. Special attention is given to the case of eigenmodes of the E symmetry of the C4v point group and those of the E1 and E2 symmetries of the C6v point group, since they are doubly degenerate on the Γ point of the Brillouin zone so that they naturally satisfy the steerability condition. We show that their dispersion curves have quadratic dependence on the wave vector in the vicinity of the Γ point. To get a linear dispersion, which is advantageous for steerable antennas, we propose a method of controlled symmetry reduction. The present theory is an extension of our previous one [Opt. Express 18, 27371 (2010)] to two-dimensional systems, for which we can achieve the deterministic degeneracy due to symmetry and the controlled symmetry reduction becomes available. This design of metamaterial steerable antennas is advantageous in the optical frequency.