# Rank-constrained least-squares solution of the Sylvester matrix equation

For the Sylvester matrix equation $AX+XB=C$, I want to find the least-squares solution $X$ via

$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{rank} (X) \leq k\end{array}$$

I searched broadly online but couldn't find any literature on this. When dealing with a rank constraint, I think that we will use the Eckart-Young Theorem somewhere. However, if we solve the Sylvester equation without the rank constraint using the Kronecker product, it's hard to deal with the rank of $X$ when $X$ is in vectorized form. I also tried to do SVD on $A$ and $B$ but could not proceed either.