This paper is devoted to general properties of the central configurations: we do not make any restriction on the number $n\geqslant$3 of particles nor on the dimension $d\geqslant1$ of the configuration. Part 7 considers however the particular case $n=d+2$, of interest because the case $n=4$, $d=2$ is the first for which we cannot solve the equations for central configurations. Our main result is Proposition 6, which gives some estimates implying an important estimate due to [23]. Our main tool is Equation $(4.5)$.

We discuss noncanonical transformations connecting different integrable systems on the symplectic leaves of the Poisson manifolds. The special class of transformations, which consists of the symplectic mappings of symplectic leaves and of the parallel transports induced by diffeomorphisms in the base of symplectic foliation, is considered. As an example, we list integrable systems associated with the spherical top. The corresponding additional integrals of motion are second, third and six order polynomials in momenta.

Linear Poisson brackets on $e(3)$ typical of rigid body dynamics are considered. All quadratic Hamiltonians of Kowalevski type having additional first integral of fourth degree are found. Quantum analogs of these Hamiltonians are listed.

In this paper we present the nonlinear Poisson structure and two first integrals in the problem on plane motion of circular cylinder and $N$ point vortices in the ideal fluid. A priori this problem is not Hamiltonian. The particular case $N = 1$, i.e. the problem on interaction of cylinder and vortex, is integrable.

We consider the problem of group classification of one non-linear partial differential equation which describes the processes on non-linear diffusion and thermal conductivity. We found Lie algebras of symmetries of this equation which let us find some exact solutions.

We consider a real analytic two degrees of freedom Hamiltonian system possessing a homoclinic orbit to a saddle-center equilibrium $p$ (two nonzero real and two nonzero imaginary eigenvalues). We take a two-parameter unfolding for such a system and show that in the case of nonresonance there are countable sets of multi-round homoclinic orbits to $p$. We also find families of periodic orbits, accumulating a the homoclinic orbits.

In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its trajectory is finit are obtained. The bifurcation diagrams are constructed.

In this paper we derive the Lie algebraic formulation of the Chen–Lee–Liu (CLL) type generalization of derivative nonlinear Schrödinger equation. We also explore its Lie algebraic connection to another derivative nonlinear Schrödinger equation, the Kaup–Newell system. Finally it is shown that the CLL equation is related to the Dodd–Caudrey–Gibbon equation after averaging over the carrier oscillation.

We investigate whether a search light, $S$, illuminating a tiny angle ("cone") with vertex $A$ inside a bounded region $Q \in \mathbb{R}^2$ with the mirror boundary $\partial Q$, will eventually illuminate the entire region $Q$. It is assumed that light rays hitting the corners of $Q$ terminate. We prove that: $(\mathbf{I})$ if $Q =$ a circle or an ellipse, then either the entire $Q$ or an annulus between two concentric circles/confocal ellipses (one of which is $\partial Q$) or the region between two confocal hyperbolas will be illuminated; $(\mathbf{II})$ if $Q =$ a square, or $(\mathbf{III})$ if $Q =$ a dispersing (Sinai) or semidespirsing billiards, then the entire region $Q$ is will be illuminated.