Unified explanation of the solar and atmospheric neutrino puzzles in a 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 $126+\overline{126}$ representation, if the $\overline{126}$ 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 Liquid Scintillation Neutrino Detector observations nor does it have neutrino hot dark matter.