In a previous paper, the author proposed Symmetry Finder (SF) method for hunting symmetries in neutrino oscillation in matter, which essentially identifies a symmetry in the diagonalized Hamiltonian in matter. It was successfully applied to Denton {it et al.} (DMP) perturbation theory to identify the eight 1-2 state exchange symmetries. In this paper, we apply the SF method to the atmospheric-resonance perturbation theory and uncover the sixteen 1-3 state exchange symmetries. Meanwhile, an alternative method for finding symmetry has been discussed. If a symmetry in the vacuum part of the Hamiltonian is found, it can be regarded as the symmetry of the total Hamiltonian because the matter term is invariant, the vacuum symmetry (VS) approach. We discuss the relationship between these two methods. One of the key questions is whether the VS method can reproduce the symmetries obtained by the SF method, to which several counter arguments are presented. Moreover, we argue that the newly found 1-3 state exchange symmetries add even more difficulties. The way how the VS method could make the goal are discussed.
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