Abstract
Three-ﬂavor neutrino oscillations in matter can be described by three eﬀective neutrino masses mi (for i = 1, 2, 3) and the eﬀective mixing matrix Vαi (for α = e, µ, τ and i = 1, 2, 3). When the matter parameter a ≡ 2√2GFNeE is taken as an independent variable, a complete set of ﬁrst-order ordinary diﬀerential equations for m2
i and |Vαi|2have been derived in the previous works. In the present paper, we point out that such a system of diﬀerential equations possesses both the continuous symmetries characterized by one-parameter Lie groups and the discrete symmetry associated with the permutations of three neutrino mass eigenstates. The implications of these symmetries for solving the diﬀerential equations and looking for diﬀerential invariants are discussed.