@Ivan So the form of the matrices $M$ and $N$ depends basically on $\det(C(z) ) \neq 0$ for all $z\in \mathbb{C} $, but what if $\det(C(z) ) =0$ for some points and $\det(C(z) ) \neq 0$ for other points? Should the matrices $M$and $N$ be independent of those $z$ where the determinant is zero or nonzero?

@IvanMeir, Thank you, brilliant work!

Is it possible to prove that this condition on $A$ and $B$ is the only one?

the equation says that $A(z) A^{*}(w)+... $, but you used $A(z) A^{*} (z) ... $ in the first line!

@ivan, that's true, thank you!