On the matrix equation XA-AX=X^p

We study the matrix equation $XA-AX=X^p$ in $M_n(K)$ for $1< p <n$. It is shown that every matrix solution $X$ is nilpotent and that the generalized eigenspaces of $A$ are $X$-invariant. For $A$ being a full Jordan block we describe how to compute all matrix solutions. Combinatorial formulas for $A^mX^{\ell},X^{\ell}A^m$ and $(AX)^{\ell}$ are given. The case $p=2$ is a special case of the algebraic Riccati equation.