Let $X,M\in M_n(\mathbb R)$ and additionally suppose that $X$ is a convergent matrix, that is to say its spectral radius $\rho(X)$ is strictly less than $1$. Define matrix $C$ to be
$$C=\sum_{k=1}^\infty X^k M (X^\top)^k.$$
Is there a closed analytic form for matrix $C$ in terms of matrices $X$ and $M$ and possibly some constant matrices? If not in general, how about for special cases such as $M=I$?
I did not get very far on my own. Let $D=M+C$, now $X D X^\top=D-M$. If one could solve for $D$ one would find $C$ as well, but I don't know how to do that. By the way, I think $C$ should exist whenever $X$ is convergent, is this correct?