Abstract
We consider an n -th order linear equation with differential operator corresponding to the direction of the main diagonal in the space of independent variables; its coefficients. in general, are variable but constant on the diagonal. We establish conditions on variable eigenvalues which give a possibility to apply some known methods of the theory of for ODEs, when integrating the equation. On this basis, the structures of solutions to the homogeneous equation are determined. We give conditions for existence of multiperiodic solutions of the equations related to variable eigenvalues and initial functions. An integral representation of multiperiodic solution to the inhomogeneous equation is given. We also suggest the concepts of variable frequency and variable period.