Related papers: Weierstrass integrability of differential equation…

This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is…

We study several aspects of the regular deformations of completely integrable systems. Namely, we prove the existence of a Hamiltonian normal form for these deformations and we show the necessary and sufficient conditions a perturbation has…

This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of…

In this paper we study the general concept of integrability in the broad sense within the frame of differential Galois theory. We concentrate on the gradient systems which are not integrable. In spite of it, if we consider them as the real…

n this paper we formulate necessary conditions for the integrability in the Jacobi sense of Newton equations $\ddot \vq=-\vF(\vq)$, where $\vq\in\C^n$ and all components of $\vF$ are polynomial and homogeneous of the same degree $l$. These…

We solve the analytic integrability problem for diferential systems in the plane whose origin is an isolated singularity and the first homogeneous component is a quadratic Lotka-Volterra type. As an application, we give the analytically…

We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…

We suggest a construction that, given a trajectorial diffeomorphism between two Hamiltonian systems, produces integrals of them. As the main example we treat geodesic equivalence of metrics. We show that the existence of a non-trivially…

Several integrability problems of differential equations are addressed by using the concept of $\mathcal{C}^{\infty}$-structure, a recent generalization of the notion of solvable structure. Specifically, the integration procedure associated…

For a large class of systems of o.d.e.'s which have first integrals, the method of arrays yields the following results: i) The first integrals $I$ can be found by solving systems of linear equations. ii) How the first integral $I$ responds…

It is shown that a linear separation relations are fundamental objects for integration by quadratures of St\"{a}ckel separable Liouville integrable systems (the so-called St\"{a}ckel systems). These relations are further employed for the…

We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra…

It is very fundamental to study irreducible plane curve singularities in algebraic geometry. The contents of the paper consist of three parts, called Part[A], Part[B] and Part[C] with Good Appendix. Our aim is to prove by Part[B] and…

We classify integrable Hamiltonian equations in 3D with the Hamiltonian operator d/dx, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality w such that w_x=u_y. Based on the method…

This article is devoted to obtain new sufficient conditions for an extremum in problems of classical calculus of variations. The concept of a set of integrands is introduced. Using this concept, first and second order sufficient conditions…

Construction and classification of 2D superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of…

Here we present/implement an algorithm to find Liouvillian first integrals of dynamical systems in the plane. In \cite{JCAM}, we have introduced the basis for the present implementation. The particular form of such systems allows reducing…

The intrinsic nature of a problem usually suggests a first suitable method to deal with it. Unfortunately, the apparent ease of application of these initial approaches may make their possible flaws seem to be inherent to the problem and…

We study algebraic integrability of complex planar polynomial vector fields $X=A (x,y)(\partial/\partial x) + B(x,y) (\partial/\partial y) $ through extensions to Hirzebruch surfaces. Using these extensions, each vector field $X$ determines…

The Fourier transform is naturally defined for integrable functrions. Otherwise, it should be stipulated in which sense the Fourier transform is understood. We consider some class of radial and, generally saying, nonintegrable functions.…