Abstract
The system of the equations with differential operator in the directions of a multiperiodic potential vector field are considered. The condition is given that determines the non-vorticity of the vector field. It is assumed that the coordinates of the vector field have the properties of periodicity and smoothness, where the periods are rationally incommensurable positive constants. The conditionally periodic structure of the characteristics is studied. The characteristics of the given differential operator are constructed, which leads to the solution of the functional equation. The case of $$m=2$$ is investigated in more detail. The results are illustrated on the specific example. Then linear and nonlinear oscillations are investigated on the vortex-free multiperiodic vector field. It is assumed that the matrix and the given vector-function satisfy the conditions of periodicity and smoothness. It is established that the linear and quasilinear systems have a unique multiperiodic solution. In conclusion, quasiperiodic oscillations generated by multiperiodic oscillations on the vortex-free vector field are investigated.