  Three-flavor neutrino oscillations in matter can be described by three
effective neutrino masses $\widetilde{m}^{}_i$ (for $i = 1, 2, 3$) and the
effective 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 ordinary
differential equations for $\widetilde{m}^2_i$ and $|V^{}_{\alpha i}|^2$ have
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.
