Integration of linear equation with differentiation operator by main diagonal of a space of independent variables with constant on it coefficients

We consider the $n$-th order linear equation with differentiation operator in the direction of the main diagonal of the space of independent variables and with variables but constants coefficients on the diagonal. The conditions on variable eigenvalues gave the possibility, when integrating the equation, to realize known methods for ordinary differential equations are established. On this basis, the structures of solutions of the homogeneous equation are determined. The conditions for existence of multiperiodic solutions of the equations related to variable eigenvalues and initial functions are given. The integral representation of a multiperiodic solution of nonhomogeneous equation is given. The concepts of variable frequency and variable period are introduced.