Fano manifolds of Picard number one whose co-tangent bundle is algebraically completely integrable system and its endomorphisms

Abstract: Let $X$ be a projective Fano manifold of Picard number one, different from the projective space. There is a folklore conjecture that any non-constant endomorphism of $X$ is an isomorphism. In the first half of this article, we will prove the folklore conjecture when the co-tangent bundle of $X$ is algebraically completely integrable system and the tangent bundle of $X$ is not nef. In the second half of the article, we will give examples of a collection of projective Fano manifolds of Picard rank one (different from the moduli space of vector bundles on algebraic curves) whose co-tangent bundles are algebraically completely integrable system. As applications of our main theorem and examples, in fact give alternative proofs of three major results appeared in three different articles.