Contents -- Preface -- Chapter 1 General Properties of Nonlinear Dynamical Systems -- 1.1 Finite-dimensional dynamical systems -- 1.1.1 Invariant measure -- 1.1.2 The Liouville condition -- 1.1.3 The Poincaré theorem -- 1.1.4 The Birkhoff-Khinchin theorem -- 1.1.5 The Birkhoff-Khinchin theorem for discrete dynamical systems -- 1.2 Poissonian and symplectic structures on manifolds -- 1.2.1 Poisson brackets -- 1.2.2 The Liouville theorem and Hamilton-Jacobi method -- 1.2.3 Dirac reduction: Symplectic and Poissonian structures on submanifolds -- Chapter 2 Geometric and Algebraic Properties of Nonlinear Dynamical Systems with Symmetry: Theory and Applications -- 2.1 The Poisson structures and Lie group actions on manifolds: Introduction -- 2.2 Lie group actions on Poisson manifolds and the orbit structure -- 2.3 The canonical reduction method on symplectic spaces and related geometric structures on principal fiber bundles -- 2.4 The form of reduced symplectic structures on cotangent spaces to Lie group manifolds and associated canonical connections -- 2.5 The geometric structure of abelian Yang-Mills type gauge field equations via the reduction method -- 2.6 The geometric structure of non-abelian Yang-Mills gauge field equations via the reduction method -- 2.7 Classical and quantum integrability -- 2.7.1 The quantization scheme, observables and Poisson manifolds -- 2.7.2 The Hopf and quantum algebras -- 2.7.3 Integrable flows related to Hopf algebras and their Poissonian representations -- 2.7.4 Casimir elements and their special properties -- 2.7.5 Poisson co-algebras and their realizations -- 2.7.6 Casimir elements and the Heisenberg-Weil algebra related structures -- 2.7.7 The Heisenberg-Weil co-algebra structure and related integrable flows.

Chapter 3 Integrability by Quadratures of Hamiltonian and Picard-Fuchs Equations: Modern Differential-Geometric Aspects -- 3.1 Introduction -- 3.2 Preliminaries -- 3.3 Integral submanifold embedding problem for an abelian Lie algebra of invariants -- 3.4 Integral submanifold embedding problem for a nonabelian Lie algebra of invariants -- 3.5 Examples -- 3.6 Existence problem for a global set of invariants -- 3.7 Additional examples -- 3.7.1 The Henon-Heiles system -- 3.7.2 A truncated four-dimensional Fokker-Planck Hamiltonian system -- Chapter 4 Infinite-dimensional Dynamical Systems -- 4.1 Preliminary remarks -- 4.2 Implectic operators and dynamical systems -- 4.3 Symmetry properties and recursion operators -- 4.4 Bäcklund transformations -- 4.5 Properties of solutions of some infinite sequences of dynamical systems -- 4.6 Integro-differential systems -- Chapter 5 Integrability Criteria for Dynamical Systems: the Gradient-Holonomic Algorithm -- 5.1 The Lax representation -- 5.1.1 Generalized eigenvalue problem -- 5.1.2 Properties of the spectral problem -- 5.1.3 Analysis of a generating function for conservation laws -- 5.2 Recursive operators and conserved quantities -- 5.2.1 Gradient-holonomic properties of the generating functional of conservation laws -- 5.2.2 Involutivity of conservation laws -- 5.3 Existence criteria for a Lax representation -- 5.3.1 The monodromy matrix and the Lax representation -- 5.3.2 The gradient-holonomic method for constructing conservation laws -- 5.3.3 Construction of compatible implectic operators -- 5.3.4 Reconstruction of the Lax operator algorithm -- 5.3.5 Asymptotic construction of recursive and implectic operators for Lax integrable dynamical systems -- 5.3.6 A small parameter method for constructing recursion and implectic operators.

5.4 The current Lie algebra on a cycle: A symmetry subalgebra of compatible bi-Hamiltonian nonlinear dynamical systems -- 5.4.1 Preliminaries -- 5.4.2 Hierarchies of symmetries and related Hamiltonian structures -- 5.4.3 A Lie-algebraic algorithm for investigating integrability -- Chapter 6 Algebraic, Analytic and Differential Geometric Aspects of Integrability for Systems on Functional Manifolds -- 6.1 A non-isospectrally Lax integrable KdV dynamical system -- 6.1.1 A non-isospectrally integrable nonlinear nonautonomous Schrödinger dynamical system -- 6.1.2 Lagrangian and Hamiltonian analysis of dynamical systems on functional manifolds: The Poisson-Dirac reduction -- 6.1.3 Remarks -- 6.2 Algebraic structure of the gradient-holonomic algorithm for Lax integrable systems -- 6.2.1 Introduction -- 6.2.2 The algebraic structure of the Lax integrable dynamical system -- 6.2.3 The periodic problem and canonical variational relationships -- 6.2.4 An integrable nonlinear dynamical system of Ito -- 6.2.5 The Benney-Kaup dynamical system -- 6.2.6 Integrability analysis of the inverse Korteweg-de Vries equation (inv KdV) -- 6.2.7 Integrability analysis of the inverse nonlinear Benney-Kaup system -- 6.3 Analysis of a Whitham type nonlocal dynamical system for a relaxing medium with spatial memory -- 6.3.1 Introduction -- 6.3.2 Lagrangian analysis -- 6.3.3 Gradient-holonomic analysis -- 6.3.4 Lax form and finite-dimensional reductions -- 6.4 A regularization scheme for a generalized Riemann hydrodynamic equation and integrability analysis -- 6.4.1 Differential-geometric integrability analysis -- 6.4.2 Bi-Hamiltonian structure and Lax representation -- 6.5 The generalized Riemann hydrodynamic regularization -- 6.5.1 Introduction -- 6.5.2 The generalized Riemann hydrodynamical equation for N = 2: Conservation laws, bi-Hamiltonian structure and Lax representation.

6.5.3 The generalized Riemann hydrodynamic equation for N = 3: Conservation laws, bi-Hamiltonian structure and Lax representation -- 6.5.4 The hierarchies of conservation laws and their analysis -- 6.5.5 Generalized Riemann hydrodynamic equation for N = 4: Conservation laws, bi-Hamiltonian structure and Lax representation -- 6.5.6 Summary conclusions -- 6.6 Differential-algebraic integrability analysis of the generalized Riemann and KdV hydrodynamic equations -- 6.6.1 Introduction -- 6.6.2 Differential-algebraic description of the Lax integrability of the generalized Riemann hydrodynamic equation for N = 3 and N = 4 -- 6.6.3 Differential-algebraic analysis of the Lax integrability of the KdV dynamical system -- 6.6.4 Summary remarks -- 6.7 Symplectic analysis of the Maxwell equations -- 6.7.1 Introduction -- 6.7.2 Symmetry properties -- 6.7.3 Dirac-Fock-Podolsky problem analysis -- 6.7.4 Symplectic reduction -- 6.8 Symplectic analysis of vortex helicity in magneto-hydrodynamics -- 6.8.1 Introduction -- 6.8.2 Symplectic and symmetry analysis -- 6.8.3 Incompressible superfluids: Symmetry analysis and conservation laws -- 6.8.4 Conclusions -- 6.9 Algebraic-analytic structure of integrability by quadratures of Abel-Riccati equations -- 6.9.1 Introduction -- 6.9.2 General differential-geometric analysis -- 6.9.3 Lie-algebraic analysis of the case n = 2 -- 6.9.4 Generalized spectral problem -- 6.9.5 Novikov-Marchenko commutator equation -- 6.9.6 Representation of the holonomy Lie algebra sl(2) -- 6.9.7 Algebraic-geometric properties of the integrable Riccati equations: The case n = 2 -- 6.9.8 Jacobi inversion and Abel transformation -- 6.9.9 Convergence of Abelian integrals -- 6.9.10 Analytical expressions for exact solutions -- 6.9.11 Abel equation integrability analysis for n = 3 -- 6.9.12 A final remark.

Chapter 7 Versal Deformations of a Dirac Operator on a Sphere and Related Dynamical Systems -- 7.1 Introduction: Diff( 1)-actions -- 7.2 Lie-algebraic structure of the A-action -- 7.3 Casimir functionals and reduction problem -- 7.4 Associated momentum map and versal deformations of the Diff( 1) action -- Chapter 8 Integrable Spatially Three-dimensional Coupled Dynamical Systems -- 8.1 Short introduction -- 8.2 Lie-algebra of Lax integrable (2+1)-dimensional dynamical systems -- 8.2.1 The Poisson bracket on the extended phase space -- 8.2.2 Hierarchies of additional symmetries -- 8.2.3 Closing remarks -- Chapter 9 Hamiltonian Analysis and Integrability of Tensor Poisson Structures and Factorized Operator Dynamical Systems -- 9.1 Problem setting -- 9.2 Factorization properties -- 9.3 Hamiltonian analysis -- 9.4 Tensor products of Poisson structures and source like factorized operator dynamical systems -- 9.5 Remarks -- Chapter 10 A Multi-dimensional Generalization of Delsarte-Lions Transmutation Operator Theory via Spectral and Differential Geometric Reduction -- 10.1 Spectral operators and generalized eigenfunctions expansions -- 10.2 Semilinear forms, generalized kernels and congruence of operators -- 10.3 Congruent kernel operators and related Delsarte transmutation maps -- 10.4 Differential-geometric structure of the Lagrangian identity and related Delsarte transmutation operators -- 10.5 The general differential-geometric and topological structure of Delsarte transmutation operators: A generalized de Rham-Hodge theory -- 10.6 A special case: Relations with Lax systems -- 10.7 Geometric and spectral theory aspects of Delsarte-Darboux binary transformations -- 10.8 The spectral structure of Delsarte-Darboux transmutation operators in multi-dimensions.

10.9 Delsarte-Darboux transmutation operators for special multi-dimensional expressions and their applications.