The global symmetry generated by K_n is a subgroup of the stringy gauge symmetry. We explore the part of the vacuum manifold related by this symmetry. Strong evidence is presented that the analytic classical solutions to the cubic string field theory found earlier in Refs. 5)[T. Takahashi and S. Tanimoto, J. High Energy Phys. 03 (2002), 033; hep-th/0202133] and 6)[I. Kishimoto and T. Takahashi, Prog. Theor. Phys. 108 (2002), 591; hep-th/0205275] are actually related by the symmetry and, therefore, all of them describe the same tachyon vacuum. Some remaining subtlety is pointed out.