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In this letter, we put forward a special neutrino mass matrix which isinvariant under a modified Friedberg-Lee (FL) transformation $\nu^{}_e \to\nu^{}_e-2\theta$ and $\nu^{}_{\mu, \tau} \to \nu^{}_{\mu, \tau}+\theta$ with$\theta$ being a space-time independent element of the Grassmann algebra.Compared to the original FL symmetry (with the transformation $\nu^{}_{e, \mu,\tau} \to \nu^{}_{e, \mu, \tau}+\theta$) which results in the TM2 neutrinomixing, the modified FL symmetry will lead us to the TM1 mixing which has abetter agreement with the experimental results. While the original FL symmetryhas to be broken in order to produce a realistic neutrino mass spectrum, themodified FL symmetry is allowed to remain intact and give us a vanishing$m^{}_1$. A combination of the FL symmetry with the $\mu$-$\tau$ reflectionsymmetry is also discussed.

A modified Friedberg-Lee symmetry for the TM1 neutrino mixing