Unified explanation of the Solar and Atmospheric neutrino Puzzles in a minimal supersymmetric SO(10) model

It was recently suggested that in a class of supersymmetric SO(10) models with Higgs multiplets in 10, and a single $126+\bar{126}$ representations, if the $\bar{126}$ contributes both to the right handed neutrino masses as well as to the charged fermion masses, one can have a complete prediction of the neutrino masses and mixings. It turns out that if one chooses only one 10, there are no regions in the parameter space where one can have a large $\nu_{\mu}-\nu_{\tau}$ mixing angle necessary to solve the atmospheric neutrino deficit while at the same time solving the solar neutrino puzzle via the $\nu_e \leftrightarrow \nu_{\mu}$ oscillation. We show that this problem can be solved in a particular class of SO(10) models with a pair of 10 multiplets if we include the additional left-handed triplet contribution to the light neutrino mass matrix. This model cannot reproduce the mass and mixing parameters required to explain the LSND observations neither does it have have a neutrino hot dark matter.