On the complete integrability of gradient systems on manifold of the beta family of the first kind (Q6558544)

This paper considers a question of complete integrability on statistical manifolds, namely Riemannian manifolds whose points are probability distributions. Here the authors examine a manifold whose points are beta distributions of the first kind with two parameters. They show the existence of a completely integrable Hamiltonian gradient system on this manifold.\N\NThe main result is that on the manifold defined by \[S = \left\{p_\theta(x) = \frac{1}{B(\theta_1 +1, \theta_2 +1)} \ x^{\theta_1} (1-x)^{\theta_2}\right\}, \] with \(\theta_1, \theta_2 \in (-1, \infty)\), \(x \in [0,1]\) and \[ B(\theta_1, \theta_2) = \int_{0}^{1} x^{\theta_1} (1-x)^{\theta_2} \,dx,\] (called by them ``dualistic manifold'') it is possible to find a potential whose associated gradient system is a completely integrable Hamiltonian system. The authors also argue that this system has two Lax pair representations defined in terms of a symmetric matrix.