This paper, it is develops an important example of the main Theorem on complete Integrability of gradient systems on a manifold admitting a potential in odd dimension in [11]. In [10] and [11], the dynamical system describes the passage in time of any object in the space of states S of the physical system. Being given a differentiable manifold S, any section X of the tangent fibred at a point p of S denoted by $TS={\cup }_{p\in S}{T}_{p}S$, induces a symmetric and positive bilinear form g called a Riemannian metric. In [7], a statistical manifold $(S,g,\nabla ,{\nabla }^{*})$, is a Riemannian manifold $(S,g)$ with a dual connection pair $(\nabla ,{\nabla }^{*})$; where g is a Riemannian manifold and ∇ is a affine connection on S and verify where $X,Y,Z$ are the vector fields. In [5], the Riemannian metric g induces an isomorphism between the module of vector fields $\mathfrak{X}\left(S\right)$ and the modulus of linear forms $\Omega \left(S\right)$ on S, denoted $b_g$. Let $\theta =\left({\theta }_1,\cdots ,{\theta }_n\right)$ a coordinate system on S. In [5] and [11], a gradient system on the Riemannian manifold is the negative flow of the gradient field $gra{d}_g\left(\mathsf{\Phi }\right)$ defined by where $\mathsf{\Phi }\in C^{\infty }\left(S\right)$ is the potential function associated with the gradient field $gra{d}_g\left(\mathsf{\Phi }\right)$ with respect to g. ${\partial }_{\theta }\mathsf{\Phi }\left(\theta \right)={\left({\partial }_{{\theta }_1}\mathsf{\Phi }\left(\theta \right),\cdots ,{\partial }_{{\theta }_n}\mathsf{\Phi }\left(\theta \right)\right)}^T$, such that $\left[b_g\right]={\left(g_{ij}\right)}_{1\le i;j\le n}$, with ${\left(g_{ij}\right)}_{1\le i;j\le n}$ is the Fisher information matrix on S. In this paper, the integral curves of the gradient system on the three-parameter bivariate beta manifold of the first kind are determined. Jacobi’s integration of the geodesics of a three-axis system of three geodesics on a three-axis ellipsoid, the motion of the vertex under the effect of gravity by Kovalevski in [2] for particular axis ratios. These efforts culminated in the work of Jacobi, who skilfully applied the method of separating variables to partial differential equations, the Hamilton-Jacobi, associated with the mechanical system in order to their integrable nature. In [1], Poincare recognized that integrability is an exceptional phenomenon of the Hamiltonian system. In 1993, Nakamura [3], proves the complete integrability of gradient systems constructed on the Gaussian and multinomial manifold. In the same year, Fujiwara [6], generalized Nakamura’s work to even dimensions. In 2023, in [11], it is studied the complete integrability of gradient systems defined on a statistical manifold in odd-dimension. How can we characterize the gradient system on S in order to study the integral curves on the three-parameter bivariate beta manifold of the first kind? Following this question, it is shown on $S=\left\{p_{\theta }\left(x\right)=\frac{1}{B(a,b,c)}{x}_1^{a-1}{x}_2^{b-1}{(1-x_1-x_2)}^{c-1},\begin{array}{cc}\theta =(a,b,c)\in {\mathbb{R}}_{+}^{*}\times {\mathbb{R}}_{+}^{*}\times {\mathbb{R}}_{+}^{*}\hfill & \hfill \\ x=(x_1,x_2)\in {\mathbb{R}}_{+}^{*}\times {\mathbb{R}}_{+}^{*}\hfill & \hfill \\ x_1+x_2<1\hfill & \hfill \\ B(a,b,c)=\frac{\mathsf{\Gamma }\left(a\right).\mathsf{\Gamma }\left(b\right).\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }(a+b+c)}\hfill & \hfill \end{array}\right\}$

where $p_{\theta }$ is the Bivariate Beta family of the first kind with three parameters, that $p_{\theta }$ is an exponential function. Therefore, according to Ovidiu [9], $\mathsf{\Phi }=-log\frac{1}{B(a,b,c)}$ is its related potential function. It is concluded that S admits a dual-coordinate pair $\left(\theta ,\eta \right)$. The gradient system is linearizable and we have to take the following form ${\dot{\eta }}_i=-{\eta }_i.$ It is proven that the associate gradient system is Hamiltonian with $\mathcal{H}$ which is in involution and satisfying $d\mathcal{H}=0.$ Therefore, applying the fundamental theorem and prove that On a statistical manifold S of three-dimensional defined by the Bivariate Beta family of the first kind with three parameters, the gradient system obtained is a sub-dynamical system of a 4-dimensional system and is a completely integrable system. From Stirling’s formula [4], it is shown that the potential function is given by: $\mathsf{\Phi }\left(\theta \right)=(a+b+c-\frac{1}{2})log(a+b+c-1)+(\frac{1}{2}-a)log(a-1)+(\frac{1}{2}-b)log(b-1)+(\frac{1}{2}-c)log(c-1)-log\left(2\pi \right)-2.$ and for all $\mathsf{{\rm Y}}=\left\{(a,b,c)\in {\mathbb{R}}^{3}0\le a\le 1,0\le b\le 1,0\le c\le 1\right\}$ the cube of edges $1cm$. We show that the dual potential function associated on S is defined for all $(a,b,c)\in {\mathbb{R}}^3-\left\{\mathsf{{\rm Y}}\right\}$ is given by, $\mathsf{\Psi }\left(\eta \right)=-\frac{a}{2(a-1)}-\frac{b}{2(b-1)}-\frac{c}{2(c-1)}+\frac{1}{2}log(a+b+c-1)-\frac{1}{2}log(a-1)-\frac{1}{2}log(b-1)-\frac{1}{2}log(c-1)-k.$ So, we prove that for all $\mathsf{{\rm Y}}$ the cube such that $\mathsf{{\rm Y}}=\left\{(a,b,c)\in {\mathbb{R}}^{3}0\le a\le 1,0\le b\le 1,0\le c\le 1\right\}$, let $(a,b,c)\in {\mathbb{R}}^3-\left\{\mathsf{{\rm Y}}\right\}$, let $D=<A(0,\frac{3}{2},\frac{3}{2});\overrightarrow{e}\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)>$ the vector line, let $\mathbb{V}=<\left(\frac{8bc-15b-15c+27}{4cb-8b-8c+15},b,c\right)>$

We also, prove that the gradient system on S obtained by using Stirling’s formula has the following linear representation ${\dot{\eta }}_i=-{\eta }_i.$ Using [11], we prove that the gradient system is Hamiltonian and completely integrable. We show that the gradient system on S has the following Lax pair representation: $\dot{L}=\left[L,N\right]$ where L is a symmetric matrix and N is the diagonal matrix. After the introduction, in Section 2, recall the preliminaries motion on theory of statistical manifold, in Section 3 we determine the Riemannian structure on Bivariate Beta family of the first kind with three parameters distribution, in Section 4, linearization and integrability of Gradient Systems, in Section 5, we determine the Riemannian structure related to Stirling’s formula. In Section 6, we have the integrability of Gradient Systems. At the end, we show that this gradient system admits a Lax pair representation.

Let $S=\left\{p_{\theta }\left(x\right),\begin{array}{cc}\theta \in \Theta \hfill & \hfill \\ x\in \mathcal{X}\hfill & \hfill \end{array}\right\}$ be the set of probabilities $p_{\theta }$, parametrized by $\Theta$, open a subset of ${\mathbb{R}}^n$; on the sample space $\mathcal{X}\subseteq \mathbb{R}$. Let $\mathcal{F}(\mathcal{X},\mathbb{R})$ be the space of real-valued smooth functions on $\mathcal{X}$. According to Ovidiu [9], the log-likelihood function is a mapping defined by Sometimes, for convenient reasons, this will be denoted by $l(x,\theta )=l\left(p_{\theta }\right)\left(x\right)$.

In [3] and [7], the Fisher information defined by

Denote $G={\left(g_{ij}\right)}_{1\le i;j\le n}$ the Fisher information matrix, the gradient system is given by The complete integrability of gradient system (3) is proven if the Theorem 1 in [11] is verify.

In this section, we present the particular properties of beta bivariate Beta family of the first kind with three parameters and associated information metric and associated gradient system.