2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79868We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Various parametric forms of trajectories are associated with different integrals of motion, Lax equations, separated variables and action-angles variables. In this review we will discuss namely these induced transformations instead of the various parametric form of the geometric objects.23 pages, LaTeX, a review for J.Nonlinear Math. PhysExactly Solvable and Integrable SystemsHigh Energy Physics - TheoryMathematical Physicstext