Nearly integrable SO(3)-structures on 5-dimensional Lie groups

The topology of Liouville foliation for the Kovalevskaya integrable case on the Lie algebra so(4)

In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra $\textrm{so}(4)$, which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank $0$ are determined, the bifurcations of Liouville tori are described and the loop molecules for all singular points of the bifurcation diagram are computed. It follows from the obtained result…

Integration of Holomorphic Lie Algebroids

We prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. As a consequence we give another proof of the following theorem: a holomorphic Poisson manifold is integrable if, and only if, its real (or imaginary) part is integrable as a real Poisson manifold.

Integrable LCK manifolds

We study a natural class of LCK manifolds that we call integrable LCK manifolds: those where the anti-Lee form $η$ corresponds to an integrable distribution. As an application we obtain a characterization of unimodular integrable LCK Lie algebras as Kähler Lie algebras equipped with suitable derivations.

Souriau-Fisher度量与在李群SO(2)和SO(3)上的完全可积系统

Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3)

本文研究了Souriau-Fisher度量在SO(2)和SO(3)上的应用，探讨了2-余积对梯度系统可积性的影响，并展示了如何通过余积修改Fisher度量。

About integrability of almost complex structures on strictly Nearly Kähler 6-manifolds

We show that any almost complex structure, positively tamed with $ω$ on nearly Kähler 6-manifold $(M,g,J,ω)$ is not integrable

Group invariance of integrable Pfaffian systems

Let $\mathcal{S}$ be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra $g$ and consequently invariant under the local action of a Lie group $G$, we show that the vertical variational cohomology of $\mathcal{S}$ is equal to the Lie algebra cohomology of $g$ with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions…

Biharmonic maps into compact Lie groups and the integrable systems

The reduction of biharmonic maps equation in terms of the Maurer-Cartan form for all smooth map of any compact Riemannian manifolds into a compact Lie group with bi-invariant Riemannian metric is obtained. By this formula, all the biharmonic curves into a compact Lie group and all biharmonic maps from a 2-dimensional open domain into a compact Lie group are characterized.

Positive metric entropy arises in some nondegenerate nearly integrable systems

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. In this paper we present a Lagrangian perturbation of the geodesic flow on a flat 3-torus. The perturbation is $C^\infty$ small but the flow has a positive measure of trajectories with positive Lyapunov exponent, namely, the flow has positive metric entropy. From this result we g…

与彩色李代数相关的超可积层次

Super integrable hierarchies associated with color Lie algebra

本文通过考虑彩色李代数 $\mathfrak{sp}_{1}(6)$ 上的两个非等谱问题，构造了 (1+1) 维和 (2+1) 维超可积系统，并利用超迹恒等式建立了其超哈密顿结构。

Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type

The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value pro…

Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.

具有简单无限维李（超）代数对称性的不可积分布

Non-integrable distributions with simple infinite-dimensional Lie (super)algebras of symmetries

本文分类了复数域上具有简单无限维李超代数对称性的不可积分布，并在特征 $p>0$ 的代数闭域上分类了已知简单向量李（超）代数的 W-分次，发现了多种新分布及其对称代数。

Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type

We prove that a local Hamiltonian operator of hydrodynamic type K_1 is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K_2 if and only if the operator K_1 is locally the Lie derivative of the operator K_2 along a vector field in the corresponding domain of local coordinates. This result gives a natural invariant definition of the class of special flat manifolds corresponding to all the class of compatible Dubrovin--Novikov Hamiltonian operators (the Frobenius--Dubrovin manifolds naturally belong to this class). There is an integrable bi-Hamiltonian hierarchy cor…

On integrability of transverse Lie-Poisson structures at nilpotent elements

We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.