  In this paper we study the algebraic properties of a new integrable
differential-difference equation. This equation can be seen as a deformation of
the modified Narita-Itoh-Bogoyavlensky equation and has the Kaup-Kupershmidt
equation in its continuous limit. Using its Lax representation we explicitly
construct a recursion operator for this equation and prove that it is a
Nijenhuis operator. Moreover, we present the bi-Hamiltonian structures for this
new equation.
