On the Integrability of a Lattice Equation with Two Continuum Limits

We study a new example of lattice equation being one of the key equations of a recent generalized symmetry classification of five-point differential-difference equations. This equation has two different continuum limits, which are the well-known fifth-order partial-differential equations, namely, the Sawada–Kotera and Kaup–-Kupershmidt equations. We justify its integrability by constructing an $L$-$A$ pair and a hierarchy of conservation laws.