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In the paper, we investigate the gauge fields that are characterized by the existence of non-trivial integrals of motion for the Wong equations. For the gauge group 𝑆𝑈(2), the class of fields admitting only the isospin first integrals is described in detail. All gauge non-equivalent Yang–Mills fields admitting a linear integral of motion for the Wong equations are classified in the three-dimensional Euclidean space

We describe the relation between operators of invariant differentiation and invariant operators on orbits of Lie group actions. We propose a new effective method for finding differential invariants and operators of invariant differentiation and present examples.

A parametric family of particular solutions to the Klein–Gordon equation with nonlocal nonlinearity in commutative homogeneous spaces is constructed with the help of the method of orbits. The nonlinear Klein–Gordon equation in the plane in which the group E(2) acts is considered.

We construct the general and N-soliton solutions of an integro-differential Schrodinger equation with a nonlocal nonlinearity. We consider integrable nonlinear integro-differential equations on the manifold of an arbitrary connected unimodular Lie group. To reduce the equations on the group to equations with a smaller number of independent variables, we use the method of orbits in the coadjoint representation and the generalized harmonic analysis based on it. We demonstrate the capacities of the algorithm with the example of the SO(3) group.

A procedure for calculation of vacuum average values of the energy-momentum tensors for a spinor field on manifolds of the Lie group is developed. To solve the formulated problem, the method of generalized harmonic analysis on the Lie groups based on the method of orbits of coadjoint representation is used.

We consider the problem of constructing scalar particle wave equations in Riemannian spaces with external gauge fields whose symmetry group is the group of motions of the Riemannian space.

In the present paper, vacuum average energy-momentum tensors are calculated on a group manifold for a real scalar field by the method of coadjoint representation orbits. An example is considered for the group SO(3). The energy-momentum tensor is calculated for a conformally equivalent metric with the help of a conformal transformation.

We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces M with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space T*M based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.

We introduce the notion of the \({\mathfrak{g}}{\mathfrak{l}(V)}\)-prolongation of Lie algebras of differential operators on homogeneous spaces. The \({\mathfrak{g}}{\mathfrak{l}(V)}\)-prolongations are topological invariants that coincide with one-dimensional cohomologies of the corresponding Lie algebras in the case where V is a homogeneous space. We apply the obtained results to the spaces S1 (the Virasoro algebra) and \({\mathbb{R}}^1 \).

A method of four-dimensional Lie group integration of the Fock–Klein equation is described in the present paper. The method is based on the formalism of co-adjoint representation orbits and on the non-commutative integration method. Right-invariant metrics being solutions of the Einstein equation are also classified for manifolds of four-dimensional Lie groups.

We study identities (functional relations between the generators of the transformation group) and also algebras of invariant operators on homogeneous spaces using the method of orbits of the coadjoint representation (coadjoint orbits). This method permits establishing the relation between these two objects and elaborating an algorithm for their construction. A classification of homogeneous spaces is introduced based on the coadjoint orbit method.

We propose an algorithm for obtaining the spectra of Casimir (Laplace) operators on Lie groups. We prove that the existence of the normal polarization associated with a linear functional on the Lie algebra is necessary and sufficient for the transition to local canonical Darboux coordinates (p, q) on the coadjoint representation orbit that is linear in the “momenta.” We show that the λ-representations of Lie algebras (which are used, in particular, in integrating differential equations) result from the quantization of the Poisson bracket on the coalgebra in canonical coordinates.

Harmonic analysis is performed for a resoluble four-dimensional Lie group, which is used to integrate linear differential equations that are not integrable within the theory of variable separation.

A quantum projection method is developed on the basis of noncommutative integration of linear differential equations and the results of M. A. Ol’shanetskii and A. M. Perelomov on the integration of classical Hamiltonian systems (projection method). The method proposed makes it possible to obtain in explicit form solutions of the quantum equations whose classical analogs can be integrated by projection. Then the semisimplicity property of the symmetry algebra of the original equation is no longer a factor. The solution basis of a Schrodinger equation with the potential of an open three-particle Tod chain is constructed as a nontrivial example.

The structure of a wave-equation symmetry algebra for massless particles is studied. It is shown that the local-symmetry algebra of massless wave equations belongs to the enveloping algebra of a conformal-group Lie algebra. It is also proven that the space of nontrivial symmetries of fixed order is irreducible under adjoint representation of the conformal algebra, and its dimension is indicated.

The article is devoted to the method of constructing exact solutions of linear differential equations, which is the quantum analog of the method of projection that A. M. Ol'shanetskii and A. P. Perelomov proposed for integration of finite-dimensional Hamiltonian systems. The method developed in the article is demonstrated on a nontrivial example.

We completely describe all local symmetries (that is, differential operators of arbitrary finite orders) of the steady-state Schrodinger equation for the hydrogen atom. This description is based on the reduction of the Schrodinger equation for an isotropic harmonic oscillator to the Schrodinger equation for the hydrogen atom, which generates the reduction of the corresponding symmetry algebras. We show that for an n-dimensional isotropic harmonic oscillator, all nontrivial local symmetry operators belong to the enveloping algebra U (su(n, C)) of the algebra su(n, C). The basis of so(4, C) consists of rotation group generators and Runge-Lenz operators.

The method of noncommutative integration for linear partial differential equations [1] is extended to the case of the so-called functional algebras for which the commutators of their generators are nonlinear functions of the same generators. The linear functions correspond to Lie algebras, whereas the quadratics are associated with the so-called quadratic algebras having wide applications in quantum field theory. A nontrivial example of integration of the Klein-Gordon equation in a curved space not allowing separation of variables is considered. A classification of four- and five-dimensional quadratic algebras is performed. A method of dimensional reduction for noncommutatively integrable many-dimensional partial differential equations is suggested. Generally, the reduced equation has a complicated functional symmetry algebra. The method permits integration of the reduced equation without the use of the explicit form of its functional algebra.

The method of noncommutative integration of linear differential equations is used to construct an exact solution of the Dirac equation, which does not presume complete separation of variables, in Stackel spaces. The Dirac equation in an external electromagnetic field is integrated by this method, using one example. The Stackel space under consideration does not enable one to solve this equation exactly within the framework of the theory of separation of variables.

A method of noncommutative integration of linear partial differential equations that is analogous to noncommutative integration of finite-dimensional Hamiltonian systems is proposed. The method is based on the concept, introduced in the paper, of a λ representation of Lie algebras. The method can be applied to the integration of the Klein-Gordon equation in Riemannian spaces of non-Stackel type (i.e., in spaces that do not admit complete separation of the variables).

The quantum analogue is considered of the classical Euler equations on the Lie algebraso (4) for the integrable case. With the help of the method of non-commutative integration (A. V. Shapovalov, I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, p. 95; No. 5, p. 33 (1991)) and the method of separation of variables, the quantum equations are reduced to the solution of a system of ordinary differential equations.

The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.

Relativistic Poincare-invariant wave equations for zero-mass and heavy particles with an arbitrary spin are constructed on the basis of special infinite-dimensional representation of the Lorentz group. The equations form a compatible system of linear differential equations for an unknown scalar function and contain spin s as a parameter (arbitrary complex number). It is also shown that the equations obtained in this way include the well-known finite-component wave equations as a special case of half-integral or integral spin.

Applying the method of noncommutative integration for linear differential equations, we build exact solutions for the Dirac equation in 4-dimensional Riemann spaces, which have a 5-parameter group of automorphisms and where the Klein-Gordon and the Dirac equations are nonintegrable using the technique of complete separation of variables.

Methods of generating exactly integrable potentials for the Schrodinger equation are consolidated within the framework of a simple construction. The Abraham-Moses method is generalized to the case of the nonstationary Schrodinger equation. An algorithm is proposed for solving the Schrodinger equation based on nonlocal symmetry operators.

By the method of non-coimmutative integration of linear differential equations proposed by the authors [Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, 95 (1991)] the Klein-Gordon and Dirac equations are integrated in four-dimensional Riemannian spaces, not admitting separation of variables.

A formal scheme of the inverse scattering method is constructed for the2 + 1 Toda chain in the class of rapidly decreasing Cauchy data. Application of the inverse scattering method to the two-dimensional infinite Toda chain was made difficult by the circumstance that this system is a (2 + 1)-dimensional object, i.e., possesses time and two spatial variables, the role of one of these being played by the chain site number. Because of this, our information about the 2 + 1 Toda chain was limited to a rich set of particular solutions of soliton type obtained in the cycle of studies by the Darboux transformation method.