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## Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems

The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in C^2 is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing–van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing–van der Pol system are constructed.

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Lienard dynamical systems $x_t=y$, $y_t=-f(x)y-g(x)$ with $\deg f<\deg g<2\deg f+1$. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Lienard dynamical systems with $\deg f<\deg g<2\deg f+1$ can have at most two distinct irreducible invariant algebraic curves simultaneously and, consequently, are not integrable with a rational first integral.

A novel method of finding and classifying irreducible invariant surfaces of non-autonomous polynomial dynamical systems in the plane is presented. The general structure of irreducible invariant surfaces and their cofactors is found. The complete set of irreducible invariant surfaces for the classical forced Duffing-van der Pol oscillator is obtained. It is proved that the forced Duffing-van der Pol oscillator possesses only one independent generalized Darboux first integral provided that a constraint on the parameters is valid. In other cases generalized Darboux first integrals do not exist. Consequently, the forced Duffing-van der Pol oscillator is not integrable with two independent generalized Darboux first integrals.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.