Let $X$ be a Fano manifold (different from $\mathbb{P}^n$) of dimension $n$ and Picard rank one. If its cotangent bundle $T^*X$ admits $n$ independent Poisson commuting regular functions then $T^*X$ is a completely integrable system with the moment map $T^*X \to \mathbb{C}^n$ defined by these functions. For example, if $X$ is the moduli space of vector bundles on curves, then $T^*X$ is completely integrable system.

Question: Is there any other example of such $X$, different from the moduli space of vector bundles ?

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