  In a previous paper, Symmetry Finder (SF) method is proposed to find the
reparametrization symmetry of the state-exchange type in neutrino oscillation
in matter. It has been applied successfully to the 1-2 state exchange symmetry
in the DMP perturbation theory, yielding the eight symmetries. In this paper,
we apply the SF method to the atmospheric-resonance perturbation theory to
uncover the 1-3 state relabeling symmetries. The pure 1-3 state symmetry takes
the unique position that it is practically impossible to formulate in vacuum
under the conventional choice of the flavor mixing matrix. In contrast, our SF
method produces the sixteen 1-3 state exchange symmetries in matter. The
relationship between the symmetries in the original (vacuum plus matter)
Hamiltonian and the ones in the diagonalized system is discussed.
