Y. Nakamura established that gradient systems defined on specific statistical manifolds, such as those associated with Gaussian and multinomial distributions, satisfy the conditions of Liouville complete integrability. Furthermore, he demonstrated that gradient flows on statistical manifolds may be linearised through the application of dual coordinates from information geometry, in a manner analogous to the action-angle coordinates employed in Hamiltonian mechanics to characterise integrable systems. We extend this line of inquiry to Souriau’s symplectic model of Information Geometry for Lie groups. Subsequently, we examine algebraic complete integrability in the sense of Adler and van Moerbeke, as well as the symplectic structure underlying Lax pairs, which can be formulated in terms of algebraic-geometric structures. This approach underlines the interplay between the analytical and group-theoretical methodologies in the study of integrable systems. Finally, we study the Adler-Kostant-Symes theorem as a principal tool for the construction of integrable systems, leveraging its capacity to establish algebraic integrability, and its link to Souriau fundamental equation of Lie Groups Thermodynamics \(\left\langle {Q,\left[ {\beta ,Z} \right]} \right\rangle + \tilde{\Theta }\left( {\beta ,Z} \right) = 0\) .