Symmetry Finder for neutrino oscillation in matter: Interplay between $\nu$SM and non-unitarity

Symmetry Finder (SF), a systematic method for finding reparametrization symmetry in neutrino oscillation in matter, has been successfully applied to the Denton {\it et al.} (DMP) and the atmospheric-resonance perturbation theories to uncover, respectively, the eight 1-2 and sixteen 1-3 state exchange symmetries. Here we apply SF to the DMP theory extended to DMP-UV by including unitarity violation (UV), a possible low-energy manifestation of physics beyond the $\nu$SM. Implementation of UV into SF yields the additional two very different constraints, which nonetheless allow remarkably consistent solutions, the eight DMP-UV symmetries. Treatment of one of the constraints, the genuine non-unitary part, leads to the key identity which entails the UV $\alpha$ parameter transformation only by rephasing, which innovates the invariance proof of the Hamiltonian. It is shown that the reparametrization symmetry as a whole distinguishes between the $\nu$SM and UV sectors, suggesting its utility for diagnostics of the extended neutrino theory. An interesting picture, the $\nu$SM - UV inter-sector communications through the phases emerges.