Charged lepton contributions to bipair neutrino mixing

The bipair neutrino mixing describes the observed solar and atmospheric mixings; however, it predicts vanishing reactor mixing angle, which is inconsistent with the observed data. We explore the ways of minimally modifying the bipair neutrino mixing by including charged lepton contributions. There are two categories of the bipair neutrino mixing which are referred to as case 1 and case 2. It turns out that, without arbitrary phases, a minimal modification is realized by just considering one $e$ - $\tau$ contribution from the charged lepton sector in the case 1. On the other hand, not only $e$ - $\tau$ contribution but also $\mu$ - $\tau$ contribution is required to realize a minimal modification in the case 2.