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This paper focuses on the introduction of right-invariant Poisson-Nijenhuis structures on Lie groupoids and their infinitesimal counterparts, also known as structures. A Poisson-Nijenhuis structure refers to a combination of a Poisson structure and a Nijenhuis structure. The paper also presents a mutual correspondence between Poisson-Nijenhuis structures on Lie algebroids and Poisson-Nijenhuis structures on their corresponding Lie groupoids, under certain conditions. This correspondence demonstrates a connection between the infinitesimal and global structures. Finally, the paper concludes with an illustrative example to enhance the understanding of the concepts introduced.

As we said in our previous work [4], the main idea of our research is to introduce a class of Lie groupoids by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid, which we called coadjoint Lie groupoids. In this paper, we will examine the relationship between structural mappings of the Lie algebroid associated to Lie groupoid and co-adjoint Lie algebroid. Also, we try to construct and define integrabiliy of Hamiltonian system on the co-adjoint Lie groupoids. In addition, we show that co-adjoint Lie groupoid associated to a symplectic Lie groupoid is a symplectic Lie groupoid., Comment: 15 pages