Modified Zee mass matrix with zero-sum condition

We 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 |Uμ3|=1/2, with the emergence of a small but non-zero Ue3. We calculate how big a value can Ue3 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 S2 permutation symmetry on our mass matrix and show how a small Ue3≠0 can emerge when this S2 permutation symmetry between the second and the third generation is broken.