How to solve the matrix equation XA-AX=f(X)

Let f be an analytic function defined on a complex domain Ω and A∈Mn(C). We assume that there exists a unique α satisfying f(α)=0. When f′(α)=0 and A is non-derogatory, we completely solve the equation XA-AX=f(X). This generalizes Burde’s results. When f′(α)≠0, we give a method to solve completely the equation XA-AX=f(X): we reduce the problem to solving a sequence of Sylvester equations. Solutions of the equation f(XA-AX)=X are also given in particular cases.