2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/176481We modify the Zee mass matrix by adding a real one parameter perturbation which is purely diagonal and trace-less. We show that in this way we can explain both solar and atmospheric neutrino oscillation data. There is a correlation between the deviation from strict maximality of $|U_{μ3}|= 1/\sqrt{2}$, with the emergence of a small but non-zero $U_{e3}$. We calculate how big a value can $U_{e3}$ get when we restrict ourselves within the allowed regions of solar and atmospheric neutrino masses and mixing angles. We also discuss the impact of a $S_2$ permutation symmetry on our mass matrix and show how a small $U_{e3} \ne 0$ can emerge when this $S_2$ permutation symmetry between the second and the third generation is broken.Version to appear in Physics Letters B. (17 pages)High Energy Physics - Phenomenologytext