The Hamiltonian formulation of Nϭ3 systems is considered in general. The most general solution of the Jacobi equation in R 3 is proposed. The form of the solution is shown to be valid also in the neighborhood of some irregular points. Compatible Poisson structures and corresponding bi-Hamiltonian systems are also discussed. Hamiltonian structures, the classification of irregular points and the corresponding reduced first order differential equations of several examples are given.

Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras A 2 , B 2 , C 2 , G 2 the complete sets of integrals in both directions are found. For the simple Lie algebras of the classical series A N , B N , C N and affine algebras of series D

We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a nonstandard Lax representation. We give several examples for Nϭ2 and Nϭ3 containing the equations of shallow water waves and its generalizations with their first two general symmetries and their recursion operators. We also discuss a reduction of Nϩ1 systems to N systems of some new equations of hydrodynamic type.

We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.In the last decade we observe that even as the number of systems of integrable nonlinear differential equations possessing nonlocal reductions is increasing, there is no one so far explaining how or where such nonlocal reductions come from. The origin of nonlocal reductions was mysterious. In this work we address to this problem. We show that those systems possessing nonlocal reductions admit discrete symmetry transformations which leave the systems invariant. A special case of discrete symmetry transformation turns out to be the nonlocal reductions of the same systems. We show this fact for NLS, mKdV, SG, DS, coupled NLS-derivative NLS, loop soliton systems, hydrodynamic type systems, and Fordy-Kulish equations, and derive all possible nonlocal reductions from the discrete symmetry transformations of these systems.

We first consider the Hamiltonian formulation of n=3 systems, in general, and show that all dynamical systems in ℝ3 are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. The construction of the Poisson structures is based on solving an associated first order linear partial differential equations. We find the Poisson structures of a dynamical system recently given by Bender [J. Phys. A: Math. Theor. 40, F793 (2007)]. Secondly, we show that all dynamical systems in Rn are locally (n-1) -Hamiltonian. We give also an algorithm, similar to the case in ℝ3, to construct a rank two Poisson structure of dynamical systems in ℝn. We give a classification of the dynamical systems with respect to the invariant functions of the vector field X→ and show that all autonomous dynamical systems in ℝn are superintegrable. © 2009 American Institute of Physics