Suppose $X$ is a Fano manifold of Picard rank one such that the cotangent bundle $T^*X$ is completely algebraically integrable system. Let $\mu: T^*X \to \mathbb{C}^n$ be the moment map, where $n$ is the dimension of $X$. It is known that a general fiber of $\mu$ is irreducible.
Question: Is it possible to impose some condition on $X$ such that $W=: \{v \in \mathbb{C}^n: \mu^{-1}(v) \text{ is reducible }\}$ has codimension $\ge 2$ ?