Most General Solution of a Matrix Equation (Arising From SVD)

Question

Suppose we have an arbitrary but known $n\times m$ complex matrix $A\in\textbf C^{n\times m}$ which therefore has an $m\times n$ conjugate transpose $A^{\dagger}\in\mathbf C^{m\times n}$. Now suppose that $X\in\textbf C^{n\times m}$ is an unknown $n\times m$ complex matrix such that: $$XA^{\dagger}A=AA^{\dagger}X$$ My question: is the power series $X=\left(\sum_{k=0}^{\infty}c_k(AA^{\dagger})^k\right)A$ for some suitable choice of complex scalars $c_k\in\textbf C$ the most general solution to this matrix equation?