Differential symmetry operators first order canonical differential equations

Background. Symmetries play an important role in mechanics and theoretical physics. The main models in these sciences are differential equations and systems of equations. Therefore, the study of symmetries of differential equations has not only theoretical, but also practical meaning. Canonical second-order differential equations are one of the basic equations of mathematical physics. The article sets the task of describing first-order differential symmetry operators of canonical equations and Lie algebras formed by such operators. Materials and methods. The introduction to this work is devoted to a brief overview of the general theory of differential substitutions of dependent variables. Such substitutions generate differential symmetry operators, and first-order operators, in particular, form Lie algebras with respect to the commutator. Paragraph 2 describes in general terms the concepts that are directly used in this article. Canonical equations and Laplace invariants are introduced. Results. In section 3, we formulate and prove a theorem on necessary and sufficient conditions under which a linear differential operator of the first order is an operator of differential symmetry of a canonical equation. In section 4, the theorem is used to describe the set of differential symmetry operators of the Euler-Poisson equations. In section 5, we establish the general form of the commutator of first-order differential symmetry operators and prove that the Lie algebra of first-order differential symmetry operators of the Euler-Poisson equations is isomorphic to the Lie algebra of second-order matrices. Section 6 contains the differential symmetry operators of canonical equations with constant coefficients, as well as canonical equations of the form. The Lie algebras of such operators turn out to be solvable four-dimensional Lie algebras with a one-dimensional center. Conclusions. The results obtained seem to be quite significant. But the main result is Theorem 1, which can be used to describe Lie algebras of differential symmetry of first-order operators in other interesting cases not covered in this paper.