A new and extremely important property of the algebraic structure of symmetries of nonlinear infinite-dimensional integrable Hamiltonian dynamical systems is described. It is shown that their invariance groups are isomorphic to a unique universal Banach Lie group of currents \(G={\mathcal S}\odot Diff(T^ n)\) on an n-dimensional torus \(T^ n\). Applications of this phenomenon to the problem of constructing general criteria of integrability of nonlinear dynamical systems of theoretical and mathematical physics are considered.