Symmetry finder: A method for hunting symmetry in neutrino oscillation

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