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The aim of this paper is to demonstrate the rich interaction between the Kowalewski-Painlev\'{e} analysis, the properties of algebraic completely integrable (a.c.i.) systems, the geometry of its Laurent series solutions, and the theory of Abelian varieties. We study the classification of metrics for which geodesic flow on the group $SO(n)$ is a.c.i. For $n=3$, the geodesic flow on $SO(3)$ is always a.c.i., and can be regarded as the Euler rigid body motion. For $n=4$, in the Adler-van Moerbeke's classification of metrics for which geodesic flow on $SO(4)$ is a.c.i., three cases come up; two are linearly equivalent to the Clebsch and Lyapunov-Steklov cases of rigid body motion in a perfect fluid, and there is a third new case namely the Kostant-Kirillov Hamiltonian flow on the dual of $so(4)$. Finally, as was shown by Haine, for $n\geq 5$ Manakov's metrics are the only left invariant diagonal metrics on $SO(n)$ for which the geodesic flow is a.c.i.

The study of KP-KdV equations are the archetype of integrable systems and are one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and a sketchy overview of this interesting subject. One of the objectives of this paper is to study the KdV equation and the inverse scattering method (based on Schrödinger and Gelfand-Levitan equations) used to solve it exactly. We study some generalities on the algebra of infinite order differential operators. The algebras of Virasoro and Heisenberg, nonlinear evolution equations such as the KdV, Boussinesq and KP play a crucial role in this study. We make a careful study of some connection between pseudo-differential operators, symplectic structures, KP hierarchy and tau functions based on the Sato-Date-Jimbo-Miwa-Kashiwara theory. A few other connections and ideas concerning the KdV and Boussinesq equations, the Gelfand-Dickey flows, the Heisenberg and Virasoro algebras are given.

We tackle in this paper the study of generalized algebraic completely integrable systems. Some interesting cases of integrable systems appear as coverings of algebraic completely integrable systems. The manifolds invariant by the complex flows are coverings of Abelian varieties and these systems are called algebraic completely integrable in the generalized sense. The later are completely integrable in the sense of Arnold-Liouville. We shall see how some algebraic completely integrable systems can be constructed from known algebraic completely integrable in the generalized sense. A large class of algebraic completely integrable systems in the generalized sense, are part of new algebraic completely integrable systems. We discuss some interesting and well known examples : a 4-dimensional algebraically integrable system in the generalized sense as part of a 5- dimensional algebraically integrable system, the H'enon-Heiles and a 5-dimensional system, the RDG potential and a 5-dimensional system, the Goryachev-Chaplygin top and a 7-dimensional system, the Lagrange top, the (generalized) Yang-Mills system and cyclic covering of Abelian varieties. [ABSTRACT FROM AUTHOR]

The aim of this paper is to realize the geodesic flow on the group SO(n) as a flow on the Kostant-Kirillov orbits. We study the adjoint and coadjoint orbits of a Lie group with an application in the case of the orthogonal special group SO(n). We will see how to explicitly determine a symplectic structure in the orbit of the coadjoint representation of a Lie group. Special attention is given to the groups SO(3) and SO(4). [ABSTRACT FROM AUTHOR]

The aim of this survey paper is to investigate the algebraic complete integrability of Euler-Arnold's body description of the four dimensional rigid body, or equivalently of geodesics in SO(4) using left-invariant metrics that arise from inertia tensors, namely non-degenerate maps Λ : so(4) → so(4)* ≡ so(4) together with the canonical inner product associated to the Killing form. Algebraic complete integrability is motivated by Arnold-Liouville's classical notion of complete integrability : one extends the value of space and time coordinates from ℝ to &8450;, and then the regular invariant manifolds are complex instead of real tori; in addition one demands such complex tori to be projective. Using different methods, as systematized by Adler-Haine-van Moerbeke-Mumford, to study the integrability of the geodesic flow on the rotation group, we will see that the linearization is carried on an abelian surface and each time a Prym variety appears related to this problem. [ABSTRACT FROM AUTHOR]

This paper presents the basic ideas and properties of elliptic functions and elliptic integrals as an expository essay. It explores some of their numerous consequences and includes applications to some problems such as the simple pendulum and the Euler rigid body motion. [ABSTRACT FROM AUTHOR]