On the complete integrability of gradient systems on manifold of the beta family of the first kind

In this article, it is shown that there exists a gradient system that is Hamiltonian and completely integrable on the beta manifold of the first kind with two parameters and whose existence depends on a potential function with duality on the manifold and which is the solution of the Legendre equation. It is shown that, by making a statistical borrowing from the gamma function, the gradient system remains Hamiltonian and completely integrable. It is shown that the gradient system constructed on the first kind of two-parameter beta manifold admits a Lax pair representation. This also makes it possible to prove its complete integrability and to determine its Hamiltonian function.