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The algebraic structure of the gradient-holonomic algorithm for Lax integrable dynamical systems is discussed. A generalization of the ℛ-structure approach for the case of operator-valued affine Lie algebras is used to prove the bi-Hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions. The monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical Lie–Poisson bracket on initial affine Lie algebra with gauge central extension. As an illustration, the two-dimensional operator Benney–Kaup integrable hierarchy is considered and their bi-Hamiltonicity is proved.

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We study periodic solutions of ordinary linear second-order differential equations with pulsed influence at fixed and nonfixed times.Numerous engineering and other practical problems are connected with the investigation of nonlinear dynamical systems with short-term processes or under the action of external forces whose duration may be neglected in creating the relevant mathematical models [1][2][3][4]. The classical example of a problem of this sort is the model of percussion clock mechanism [5]. Similar problems appear in many other fields of science and engineering, in particular, in metallurgy (in controlling temperature in thermal and open-hearth furnaces [6]), in chemical technology [7], in the dynamics of aircrafts [8], in mathematical economics [9], in medicine and biology [10], etc.The investigation of mathematical models for these systems leads to the necessity of the analysis of discontinuous dynamical systems generated by ordinary differential equations and conditions of pulsed action [4, 1 1J.The general characteristics of the qualitative behavior of solutions of systems of ordinary differential equations with pulsed influence, the similarity and difference between the problems in this field of applied mathematics and the corresponding problems in the theory of ordinary differential equations, as well as the main results obtained in this field are presented in [4,12]. Fundamental results concerning the applicability of asymptotic methods of nonlinear mechanics to weakly nonlinear differential equations with pulsed action can be found in [ 1,[13][14][15][16].In applying asymptotic methods to the investigation of weakly nonlinear differential equations, it is very important to study the so-called nonperturbed (generating) problem obtained in the case where the value of the corresponding small parameter is equal to zero. A nonlinear dynamical system generated by a linear ordinary differential equation of the form d2x dx + + qx = 0, p,q~ R,(1) d t 2 P -dtt and some conditions of pulsed action [4] can be regarded as a nonperturbed problem of this sort appearing as a result of asymptotic integration of weakly nonlinear differential equations with pulsed action [ 17,18]. There are several types of conditions of pulse action [4], among which one most often encounters conditions specified at fixed and nonfixed instants of time. In the general case, problems with pulsed action at nonfixed times are more complicated than problems with pulsed action at fixed instants of time. Mathematically, the conditions of pulsed action mean that there are certain rules according to which moving points of the considered dynamical system equation at the instants of pulsed action instantaneously changes its path by passing from one trajectory of the dynamical system to another. At first sight, this problem seems to be quite simple, but the behavior of trajectories in a dynamical system generated by ordinary differential equations with some conditions of pulsed action can be extremely complicated precisely due to the presence of...