  Symmetry in neutrino oscillation serves for a better understanding of the
physical properties of the phenomenon. We present a systematic way of finding
symmetry in neutrino oscillation, which we call Symmetry Finder (SF). By
extending the known framework in vacuum into a matter environment, we derive
the SF equation, a powerful machinery for identifying symmetry in the system.
After learning lessons on symmetry in the Zaglauer-Schwarzer system with a
matter equivalent of the vacuum symmetry, we apply the SF method to the Denton
et al. (DMP) perturbation theory to first order. We show that the method is so
powerful that we uncover the eight reparametrization symmetries with the $1
\leftrightarrow 2$ state exchange in DMP, denoted as IA, IB, $\cdot \cdot
\cdot$, IVB, all new except for IA. The transformations consist of the both
fundamental and dynamical variables, indicating their equal importance. It is
also shown that all the symmetries discussed in this paper can be understood as
the Hamiltonian symmetries, which ensures their all-order validity and
applicability to varying density matter.
