  The problem of light propagation of frequency corresponding to half of the
energy difference between a metastable excited state and the ground state of
atoms is examined, and solved for coherent medium by analytic means. We
demonstrate that the non-linear system of Maxwell-Bloch equation for the
effective model of the $\Lambda-$type three levels is integrable in the
mathematical sense. Analytic solutions thus obtained describe pulse splitting
accompanied by compression, indicating a kind of non-linear instability of
propagating pulses. The instability is eventually terminated by coherent two
photon emission (called paired superradiance or PSR in short). These results
are displayed by numerical outputs for visual understanding, as well. It is
further shown that the integrable system allows a new class of soliton
solutions. Solitons, implying the phenomenon of seff-induced transparancy at
non-resonant frequencies, are stable against PSR. One of our goals of the
present work is construction of a calculable theoretical framework for PSR
rates associated with a trigger pulse propagation, which is achieved by
combining analytic results with perturbative methods. PSR photon spectrum and
its rate $\propto$(target number density)$^2$, along with their time structure,
are clarified this way. These results may open a new path for interesting
technological applications such as quantum entanglement and for solving the
remaining problems of the still mysterious neutrino. Some basic strategy for
realistic experiments of PSR detection and soliton production is also outlined.
