  Let $K$ denote an algebraically closed field with characteristic 0, and let
$q$ denote a nonzero scalar in $K$ that is not a root of unity. Let $A_q$
denote the unital associative $K$-algebra defined by generators $x,y$ and
relations x^3y-[3]_q x^2yx +[3]_q xyx^2 -yx^3=0, y^3x-[3]_q y^2xy +[3]_q yxy^2
-xy^3=0, where $[3]_q = (q^3-q^{-3})/(q-q^{-1})$. We classify up to isomorphism
the finite-dimensional irreducible $A_q$-modules on which neither of $x,y$ is
nilpotent. We discuss how these modules are related to tridiagonal pairs.
