The authors consider the solution of systems of linear equations with special Toeplitz matrices of classes \(T\) or \(B^ 0\); the latter contains the matrices arising from the finite difference discretization of the differential operator \(\partial^{2m}/\partial x^{2m}\) over an interval with homogeneous conditions on the derivatives of lower order. Using an information about the eigenvalues and eigenvectors of such matrices, the authors introduce a class of optimal multigrid methods for the solution of such systems. Numerical experiments confirm the efficiency of the proposed methods.