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Let $X$ be a projective Fano manifold of Picard number one, different fromthe projective space. There is a folklore conjecture that any non-constantendomorphism of $X$ is an isomorphism. In the first half of this article, wewill prove the folklore conjecture when the co-tangent bundle of $X$ isalgebraically completely integrable system and the tangent bundle of $X$ is notnef. In the second half of the article, we will give examples of a collectionof projective Fano manifolds of Picard rank one (different from the modulispace of vector bundles on algebraic curves) whose co-tangent bundles arealgebraically completely integrable system.  As applications of our main theorem and examples, in fact give alternativeproofs of three major results appeared in three different articles.

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