Bi-Maximal Oscillation Solutions for Solar and Atmospheric Neutrinos

We show that if the solar and atmospheric data are both described by maximal vacuum oscillations at the relevant mass scales then there exists a unique mixing matrix for three neutrino flavors. The solution necessarily conserves CP and automatically implies that there is no disappearance of atmospheric $\nu_e$, consistent with indications from the Super-Kamiokande experiment. We also investigate the consequences for three-neutrino mixing if the solar and atmospheric oscillations exhibit mixing that is large but not maximal. For non-maximal mixing $\nu_e\leftrightarrow\nu_\tau$ and $\nu_e\leftrightarrow\nu_\mu$ oscillations are predicted that may be observable in future long-baseline experiments.