Let f be an analytic function defined on a complex domain Omega and A be a
(n,n) complex matrix. We assume that there exists a unique alpha satisfying
f(alpha)=0. When f'(alpha)=0 and A is non derogatory, we solve completely the
equation XA-AX=f(X). This generalizes Burde's results. When f'(alpha) is not
zero, we give a method to solve completely the equation XA-AX=f(X): we reduce
the problem to solve a sequence of Sylvester equations. Solutions of the
equation f(XA-AX)=X are also given in particular cases.