Continuous and discrete symmetries of renormalization group equations for neutrino oscillations in matter

Three-flavor neutrino oscillations in matter can be described by three effective masses $\widetilde{m}^{}_i$ (for $i = 1, 2, 3$) and the mixing matrix $V^{}_{\alpha i}$ $\alpha e, \mu, \tau$ 3$). When parameter $a \equiv 2\sqrt{2} G^{}_{\rm F} N^{}_e E$ is taken as an independent variable, a complete set of first-order ordinary differential equations for $\widetilde{m}^2_i$ $|V^{}_{\alpha i}|^2$ have been derived previous works. In present paper, we point out that such system possesses both continuous symmetries characterized one-parameter Lie groups discrete symmetry associated with permutations mass eigenstates. The implications these solving looking invariants are discussed.