Commuting maps on rank-$k$ matrices

Let $n\ge 2$ be a natural number. Let $M_n(\mathbb{K})$ be the ring of all $n \times n$matricesoverafield$$\mathbb{K}$$\mathrm{.}Fixnaturalnumber$k$satisfying$1<k\leq n$\mathrm{.}Underamildtechnicalassumptionover$$\mathbb{K}$$ we will show that additive maps $G:M_n(\mathbb{K})\to M_n(\mathbb{K})$ such that $[G(x),x]=0$ for every rank-$k$ matrix $x\in M_n(\mathbb{K})$ are of form $\lambda x+\mu (x)$, where $\lambda \in Z$, $\mu :M_n(\mathbb{K})\to Z$, and $Z$ stands for the center of $M_n(\mathbb{K})$. Furthermore, we shall see an example that there are additive maps such that $[G(x),x]=0$ for all rank-1 matrices that are not of the form $\lambda x+\mu (x)$. We will also discuss the $m$-additive case.