In the theory of multiperiodic systems of equations with an operator D, the necessary and sufficient condition for the multiperiodicity of their solutions is important. But such a condition is represented in the form of systems of equations in finite differences considered in the space of smooth multiperiodic functions. To solve problems of this nature, the use of well-known methods of functional analysis is problematic.

The main obstacle in this matter is the absence of periodic characteristics of the operator D in Euclidean space with Cartesian coordinates. Indeed, when we move from the initial problem for a system with an operator D to an equivalent integral system, we have to consider it along the characteristics of the differentiation operator. Then the corresponding integrand function ceases to be multiperiodic due to non-periodic characteristics when the unknown function belongs to the class of multiperiodic functions. A way out of this situation is possible if we change the vector field of the differentiation operator to a more complex geometric formation that ensures the periodicity of the characteristics. In this regard, in this paper, the following are considered as the vector field of the operator D: a circle, a torus and an infinite cylindrical surface, where the question of the periodicity of characteristics is studied. Obviously, the final choice of the variety depends on the nature of the problems for systems with a differentiation operator D. Looking ahead, we note that for our purposes it is advisable to consider systems with this operator on a cylindrical surface. This study presents the main necessary properties of periodic characteristics concerning transformations of a one-parameter family of a group. The main issues are thoroughly studied for the two-dimensional case of characteristic systems, and then extended to the multidimensional case. For the integration of multiperiodic functions, it is important to consider the unfolding of helical lines on a plane, which is given some attention. Sufficient conditions for the multiperiodicity of the characteristics of a more general differentiation operator in the directions of constant vectors are also established.