Determination of the Integral Curves of the Completely Integrable Gradient System on the Three-Parameter Beta Bivariate Statistical Manifold of the First Kind

In this paper, it is shown that the bivariate beta statistical manifold of the first kind with three parameters has a gradient system Hamiltonian and is completely integrable. It is shown that this system admits a Lax pair representation. The question here is how to construct a gradient system and show that it is completely integrable while determining the integrable curves on the three-parameter bivariate beta manifold of the first kind admitting a potential function?To do this, we prove the existence of the potential on the manifold using ovidiu in Ovidiu, then using Amari's theorems in Amari, we show that this coordinate system admits a dual coordinate system. This will allow us to determine the Riemannian metric on the variety and to construct the gradient system. We linearise this system using Nakamura's method. We prove that it is Hamiltonian and completely integrable using the theorem in \cite{mama-proceeding}. It is shown that the Bivariate Beta of the first kind with three parameters, is an exponential function. The gradient system is linearizable. It is proven that the associate gradient system is Hamiltonian with $ \mathcal{H} $ which is in involution and satisfying $d\mathcal{H}=0.$ Therefore, the gradient system obtained is a sub-dynamical system of a $4$-dimensional system and is a completely integrable system. We show that the gradient system on the statistical model has the following Lax pair representation: $ \dot{L}= \left[L, N\right] $ where $L$ is a symmetric matrix and $N$ is the diagonal matrix. The gradient system defined by the Bivariate Beta family of the first kind with three parameters odd-dimensional manifold is a completely integrable Hamiltonian system.