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

Continuous and Discrete Symmetries of Renormalization Group Equations for Neutrino Oscillations in Matter