Third-order integrable difference equations generated by a pair of   second-order equations
  We show that the third-order difference equations proposed by Hirota,   Kimura and Yahagi are generated by a pair of second-order difference equations. In some cases, the pair of the second-order equations are equivalent to the Quispel-Robert-Thomson(QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the equations. 
Junta Matsukidaira and Daisuke Takahashi