Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$ $$ A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}. $$

I'm interested in finding a factorization of $A(x)$ of the form $$\tag{$\star$} \label{fact} A(x) = C^\top(x^{-1})\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}C(x), $$ where $C(x)$ is a suitable $2\times 2$ rational matrix-valued function and $\bullet^\top$ denotes transposition. An example of such a factorization is given, for instance, by $$ C(x)=\begin{bmatrix} 1 & 0 \\ 0 & x\end{bmatrix}. $$

My question.Does there exist a factorization of $A(x)$ as in \eqref{fact} such that the factor $C(x)$ possesses the two additional properties below?

- The entries of $C(x)$ have no singularities at $x\in\mathbb{C}$, $|x|\le 1$, and
- $C(x)$ has full rank for every $x\in\mathbb{C}$ such that $|x|\le 1$.

I can find factors $C(x)$ that satisfy either property 1 or property 2, but not both. Thus, any comment/suggestion is really appreciated.