Symmetry in Neutrino Oscillation in Matter with Non-unitarity

Recently we have developed a method called ``Symmetry Finder'' (SF) for hunting the reparametrization symmetry in three-neutrino system matter. Here, apply SF to Denton {\it et al.} (DMP) perturbation theory extended by including unitarity violation (UV), possible low-energy manifestation of physics beyond $\nu$SM. Implementation UV into framework yields additional two very different constraints, which nonetheless allow remarkably consistent solutions, eight DMP-UV symmetries. Treatment one genuine non-unitary part, leads key identity entails $\alpha$ parameter transformation only rephasing, innovates invariance proof Hamiltonian. The quantum mechanical nature dictates both $\nu$SM and variables transform jointly, through response sectors are related reveal their interplay. Thus, can serve tool diagnostics, probing interrelation between description new physics. Problem vacuum is revisited complete symmetries akin DMP's.