This volume contains five surveys on dynamical systems. The first one deals with nonholonomic mechanics and gives an updated and systematic treatment of the geometry of distributions and of variational problems with nonintegrable constraints. The modern language of differential geometry used throughout the survey allows for a clear and unified exposition of the earlier work on nonholonomic problems. There is a detailed discussion of the dynamical properties of the nonholonomic geodesic flow and of various related concepts， such as nonholonomic exponential mapping， nonholonomic sphere， etc.
　　Other surveys treatvarious aspects of integrable Hamiltonian systems， with an emphasis on Lie-algebraic constructions. Among the topics covered are： the generalized Calogero-Moser systems based on root systems of simple Lie algebras， a general r-matrix scheme for constructing integrable systems and Lax pairs， links with finite-gap integration theory， topological aspects ofintegrable systems， integrable tops， etc. One of the surveys gives a thorough analysis of a family of quantum integrable systems（Toda lattices）using the machinery of representation theory.
　　Readers will find all the new differential geometric and Liealgebraic methods which are currently used in the theory of integrable systems in this book. It will be indispensable to graduate students and researchers in mathematics and theoretical physics.

Introduction

Chapter 1． Geometry of Distributions
1． Distributions and Related Objects
1．1． Distributions and Differential Systems
1．2． Frobenius Theorem and the Flag of a Distribution
1．3． Codistributions and Pfaffian Systems
1．4． Regular Distributions
1．5． Distributions Invariant with Respect to Group Actions and Some Canonical Examples
1．6． Connections as Distributions
1．7． A Classification of Left Invariant Contact Structures on Three-Dimensional Lie Groups
2． Generic Distributions and Sets of Vector Fields． and Degeneracies of Small Codimension． Nilpotentization and Classification Problem
2．1． Generic Distributions
2．2． Normal Forms of Jets of Basic Vector Fields of a GenericDistribution
2．3． Dcgeneracies of Small Codimension
2．4． Generic Sets of Vector Fields
2．5． Small Codimension Degeneracies of Sets of Vector Fields
2．6． Projection Map Associated with a Distribution
2．7． Classificaton of Regular Distributions
2．8． Nilpotentization and Nilpotent Calculus

Chapter 2． Basic Theory of Nonholonomic Riemannian Manifolds
1． General Nonholonomic Variational Problem and the Geodesic
Flow on Nonholonomic Riemannian Manifolds
1．1． Rashevsky-Chow Theorem and Nonholonomic Riemannian
Metrics （Carnot-Caratheodory Metrics）
1．2． Two-Point Problem and the Hopf-Rinow Theorem
1．3． The Cauchy Problem and the Nonholonomic Geodesic Flow
1．4． The Euler-Lagrange Equations in Invariant Form and in the
Orthogonal Moving Frame and Nonholonomic Geodesics
1．5． The Standard Form of Equations of Nonholonomic Geodesics
for Generic Distributions
1．6． Nonholonomic Exponential Mapping and the Wave Front
1．7． The Action Functional
2． Estimates of the Accessibility Set
2．1． The Parallelotope Theorem
2．2． Polysystems and Finslerian Metrics
2．3． Theorem on the Leading Term
2．4． Estimates of Generic Nonholonomic Metrics on Compact Manifolds
2．5． Hausdorff Dimension of Nonholonomic Riemannian Manifolds
2．6． The Nonholonomic Ballin the Heisenbcrg Group as the Limit of Powers of a Riemannian Ball

Chapter 3． Nonholonomic Variational Problems onThree-Dimensional Lie Groups
1． The Nonholonomic E-Sphere and the Wave Front
1．1． Reduction of the Nonholonomic Geodesic Flow
1．2． Metric Tensors on Three-Dimensional Nonholonomic Lie Algebras
1．3． Structure Constants of Three-Dimensional Nonholonomic Lie Algebras
1．4． Normal Forms of Equations of Nonholonomic Geodesics on Three-Dimensional Lie Groups
1．5． The Flow on the Base V + V1 of the Semidirect Product
1．6． Wave Front of Nonholonomic Geodesic Flow． Nonholonomic ε-Sphere and their Singularities
1．7． Metric Structure of the Sphere SeV
2． Nonholonomic Geodesic Flow on Three-Dimensional Lie Groups
2．1． The Monodromy Maps
2．2． Nonholonomic Geodesic Flow on SO（3）
2．3． NG-Flow on Compact Homogeneous Spaces of the Heisenberg Group
2．4． Nonholonomic Geodesic Flows on Compact Homogeneous Spaces of SL2R

References
Additional Bibliographical Notes