Bilinear maps on the ring of strictly upper triangular matrices

Let $R$ be a 2-torsion free unital ring and $N_n=N_n(R)$ the ring of strictly upper triangular matrices with entries in $R$ and center $Z=Z(N_n)$. It has been previously shown that any linear map $f:N_n\to N_n$ satisfying the condition $[f(X),X]=0$ must be of the form $f(X)=\lambda X+\mu (X)$ for some $\lambda \in R$ and additive map $\mu$ defined on $N_n$. We extend these known results by providing a complete description of the bilinear maps $f:N_n\times N_n\to N_n$ satisfying the identity $[f(X,X),X]=0$ for all $X\in N_n$.