Related papers: Geometry of integrable dynamical systems on 2-dime…
We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of…
A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional…
Near-integrability is usually associated with smooth small perturbations of smooth integrable systems. Studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provides an…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
We construct hierarchies of integrable systems invariant under the two-dimensional Darboux-Toda mapping for noncommuting objects, thus generalizing to the noncommutative case the integrable mapping approach to nonlinear dynamical systems.…
We investigate integrable 2-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of…
We consider some differential geometric classes of local and nonlocal Poisson and symplectic structures on loop spaces of smooth manifolds which give natural Hamiltonian and multihamiltonian representations for some important nonlinear…
We consider set of functions on Poisson manifold related by continues one-parameter group of transformations. Class of vector fields that produce involutive families of functions is investigated and relationship between these vector fields…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
The paper deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. More precisely, we study degree $2$ invariants with coefficients $\mathbb{Q}/\mathbb{Z}(1)$, that is invariants…
A novel class of integrable surfaces is recorded. This class of O surfaces is shown to include and generalize classical surfaces such as isothermic, constant mean curvature, minimal, `linear' Weingarten, Guichard and Petot surfaces and…
Some of the most outstanding questions in the field of gravitation and geometry remain unsolved as a result of our limited understanding of the global structure of the spacetime geometry and the role played by global spacetime…
In these notes we study left-invariant involutive structures on $\mathrm{SU}(2)$, the most na\"ive non-commutative compact Lie group. We determine closedness of the range (in the smooth topology) of a single complex vector field spanning…
The paper investigates invariants of compactified Picard modular surfaces by principal congruence subgroups of Picard modular groups. The applications to the surface classification and modular forms are discussed.
Invariant minimal surfaces in the real special linear group of degree 2 with canonical Riemannian and Lorentzian metrics are studied. Constant mean curvature surfaces with vertically harmonic Gau{\ss} map are classified.
We show that the space of orthogonally separable coordinates on the sphere $S^3$ induces a natural family of integrable systems, which after symplectic reduction leads to a family of integrable systems on $S^2 \times S^2$. The generic…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson…
We study the integrability of two-dimensional theories that are obtained by a dimensional reduction of certain four-dimensional gravitational theories describing the coupling of Maxwell fields and neutral scalar fields to gravity in the…