CONTENTS -- Foreword -- Acknowledgments -- FEATURED CONTRIBUTIONS -- D. Alekseevsky & A. Spiro Homogeneous bi-lagrangian manifolds and invariant Monge- Ampere equations -- 1. Bi-Lagrangian manifolds -- 2. Invariant Bi-Lagrangian structures on adjoint orbits of semisimple groups and gradations of semi-simple Lie algebras -- 2.1. Homogeneous symplectic manifolds of a semisimple Lie group -- 2.2. Fundamental gradations of a (complex or real) semisimple Lie algebra g -- 2.3. Examples of fundamental gradations -- 2.3.1. Fundamental gradations of 5K(V) -- 2.3.2. Fundamental gradations of g2 -- 3. Generalized Monge- AmpBre equations and effective forms on contact manifolds -- 4. Invariant Monge-AmpBre equations on homogeneous bi-Lagrangian manifolds -- 4.1. Invariant effective forms on contactifications of homogeneous symplectic manifolds -- 4.2. Invariant effective forms o n homogeneous bi-Lagmngian manifolds -- References -- I.M. Anderson, M.E. Fels & P.J. Vassiliou On Darboux integrability -- 1. Introduction -- 2. Pfaffian Systems and Superposition -- 3. Darboux Integrability and Superposition -- Acknowledgments -- References -- E. Asadi & J.A. Sanders Integrable systems in symplectic geometry -- 1. Introduction -- 2. Moving frame method in symplectic geometry -- 3. Cartan structure equation in symplectic geometry -- 4. The geometric operators in terms of Lie bracket, Killing form and projections -- 5. Reduction to scalar part -- Acknowledgments -- References -- V. Barutello & S. Terracini Sangularitaes and collisions in N-body type problems -- Locally minimal and generalized solutions -- Asymptotic analysis along a total collision trajectory -- Partial collisions -- Blow-up and averaging estimates -- References -- S. Benenti Computing curvature without Christoflel symbols -- 1. The six steps of the aIgorithm -- 2. Explanation -- 3. An example.

3.1. The computation of the quadratic forms Qi -- 3.2. The computation of the cubic forms Ri -- 3.3. The computation of the A-symbols -- 3.4. The computation of the B-symbols -- 3.5. The Ricci tensor -- 4. How to use this example -- References -- H. Broer, R. van Dijk & R. Vitolo periodically driven oscillators for ! = 1,2,3 Survey of strong nomnal-internal Ic : e resonances in quasi- -- 1. Introduction to the problem -- 2 . Results -- 2.1. The k : 1 resonance -- 2.2. The k : 2 resonance -- 2.3. The k : 3 resonance -- References -- A. Celletti & L. Chierchia Quasi-periodic attractors and spin-orbit resonances -- 1. Introduction -- 2. The spin-orbit model -- 3. Results -- 4. Mercury's capture -- References -- A. Degasperis & S. Lombard0 nonvanishing boundary values Darboux construction of solutions of integrable PDEs with -- 1. Introduction -- 2. Darboux matrix trasformation and integrability -- 3. Darboux transformations for coupled wave equations -- 3.1. Coupled wave equations -- 3.2. Darboux transformations -- Acknowledgments -- References -- D. Ferrario Transitive decomposition of n-body symmetry groups -- 1. Introduction -- 2. Symmetries -- 3. Reductions -- 4. Minimizers -- 5 . Examples -- 6. Conclusions -- References -- T. Gramchev & M. Yoshino fixed point Normal forms for commuting vector fields near a common -- 1. Introduction and the set-up of the problem -- 2. Convergent normal form for commuting singular vector fields -- 3. Gevrey estimates for divergent solutions of linear homological equations -- 4. Divergent NFT and centralizers for resonant vector fields -- Acknowledgments -- References -- A monkey saddle in rigid body dynamics H. Hanssmann -- 1. Introduction -- 2. The mass distribution -- 3. The normal form -- 4. Perturbation analysis -- References -- J. Krashil'shchik Nonlocal geometry of PDEs and integrability -- Introduction.

1. Natural coverings -- 2. Integrability structures -- 3. Nonlocal structures -- 4. Example: the dispersionless Boussinesq equation, Ref. [6] -- Acknowledgments -- References -- B. Kruglikov Anomaly of linearization and auxiliary integrals -- Introduction -- 1. Anomaly via Hessian -- 2. Coordinate expressions -- 3. Auxiliary integrals -- 4. Symmetries and compatibility -- 5. Conclusion -- References -- D. Krupka Natural variational principles -- 1. Introduction -- 2. Differential invariants -- 3. Global first variation formula -- 4. Natural Lagrange structures -- 5 . Example: The Hilbert Lagrangian -- Acknowledgments -- References -- 0. Krupkova Variational exterior differential systems -- 1. Introduction -- 2. Variational systems in fibred manifolds -- 3. Exterior differential systems for variational equations -- 4. Symmetries and conservation laws -- References -- S. Marmi & P. Tempesta hyperfunctions O n the relation between formal groups, Appell polynomials and -- 1. Introduction -- 2. Bernoulli-type polynomials and numbers of the first kind -- 3. Basics on hyperfunctions -- 4. Delta rational functions and polylogarithms -- References -- P. Morando Variational A-symmetries and deformed Lie derivatives -- Introduction -- 1. Preliminaries -- 2. The deformed operators dp and L p -- 3. Application to A-symmetries -- 4. Variational A-symmetries -- Acknowledgments -- References -- 0.I. Morozov Maurer-Cartan forms for symmetry pseudo-groups and coverings of differential equations -- 1. Introduction -- 2. Structure of Symmetry Pseudo-Group and a Covering for the Modified Khokhlov-Zabolotskaya Equation -- 3. Multi-Valued Solutions of the Khokhlov-Zabolotskaya Equation -- 4. Conclusion -- References -- N.N. Nekhoroshev Fuzzy fractional monodromy -- 1. General conception of the considered monodromy -- 2. Fractional monodromy and its explanation.

3. Explanation of the fuzziness of monodromy -- 4. Formulation of the result -- References -- F. Oliveri, G . Manno & R. Vitolo O n the correspondence between differential equations and symmetry algebras -- 1. Introduction -- 2. Theoretical setting -- 3. DEs characterized by their Lie point symmetries -- 4. DEs which are uniquely determined by Lie algebras of vector fields on Rnfm -- Acknowledgments -- References -- P.J. Olver & J. Pohjanpelto Moving frames and differential invariants for Lie pseudo-groups -- 1. Introduction -- 2. The Diffeomorphism Pseudo-Group -- 3. Lie Pseudo-Groups -- 4. Pseudo-Group Actions on Extended Jet Bundles -- 5. Recurrence Formulae -- 6. The Symbol Modules -- Acknowledgments -- References -- M. Procesi Periodic solutions for a class of non-linear Schroedinger equations in D > 1 spatial dimensions -- 1. Introduction -- 2. Main results -- 2.1. A sketch of the proof -- References -- M.A. Teixeira & P.R. Da Silva equations Singular perturbation for discontinuous ordinary differential -- 1. Introduction -- 2. Preliminaries and statement of the main results -- 3. Proof of Theorem 2.1 -- 4. Proof of Theorems 2.2 and 2.3 -- References -- A. Vanderbauwhede reversible systems Continuation of periodic solutions in conservative and -- 1. Introduction -- 2. Continuation of zeros of constrained mappings -- 3. Periodic orbits in conservative systems -- 4. Reversible systems -- 5 . Application to N-body problems -- References -- F. Verhulst Emergence of slow manifolds in nonlinear wave equations -- 1. Singular perturbations and slow manifolds -- 2. Extension to PDEs -- 3. Formulation for wave equations -- 4. A nonlinear Rayleigh wave equation -- 5 . Discussion -- References -- P. Winternitz & i. Yurdqen euclidean spaces Superintegrable systems with spin in two- and three-dimensional -- 1. Introduction -- 2. The Two-Dimensional Case.

3. The Three-Dimensional Case -- 4. Conclusions -- Acknowledgments -- References -- B. Zhilinskii manifestations Generalization of Hamiltonian monodromy. Quantum -- 1. Introduction -- 2. Singularities of energy-momentum maps and monodromy -- 3. Bidromy -- 4. Conclusions -- References -- EXTENDED ABSTRACTS -- M.B. Abd-el-Malek & H.S. Hassan analysis Internal flow through a conducting thin duct via symmetry -- 1. Mathematical formulation of the problem -- 2. Solution of the Problem -- 3. Results -- R. Alonso Blanco, G . Manno & F. Pugliese Contact geometry of parabolic Monge-Ampdre equations -- References -- K. Andriopoulos Complete symmetry groups and Lie remarkability -- References -- M. Arminjon Quantum wave equations in curved space-time wave mechanics -- References -- D. Catalano Ferraioli Nonlocal interpretation of X-symmetries -- 1. Introduction -- 2. Reconstruction of nonlocal symmetries from -symmet ries -- References -- C. Chanu, M. Chanachowicz & R.G. McLenaghan R-separation for the conformal Laplacian -- 1. The conformally invariant (CI) Laplace equation -- References -- A.M. Cherubini, G. Metafune & F. Paparella Asymptotic behavior of a bouncing ball -- References -- G.M. Coclite & K.H. Karlsen Discontinuous solutions for the Degasperis-Procesi equation -- References -- V. Golovko Variational Poisson-Nijenhuis structures for evolution PDEs -- 1. Main result -- Acknowledgments -- References -- A. Gonzalez-Enriquez & R. de la Llave Analytic approximations of geometric maps and applications to K A M theory -- References -- M. Iwasa A method to construct asymptotic solutions invariant under the renormalization group -- 1. The Renormalizatoin group method with Lie symmetry -- 2. Concluding remarks -- References -- J. JanySka Utiyama's reduction method and infinitesimal symmetries of invariant Lagrangians -- Acknowledgment -- References.

I.I. Kosenko & S.Ya. Stepanov Stability of the tethered satellite system relative equilibria. Unrestricted problem.