Summary: Two-sided Ostrowski matrix iteration is applied to solve eigenproblems of non-conservative continuous systems in structural dynamics starting from known eigensolutions of a related conservative system. Eigensolution paths are followed as the parameters of the system governing its non- conservative behaviour are increased from zero. No discretization is performed and no truncation errors are introduced. The real eigenvalues of the related conservative problem are calculated by use of the Wittrick-Williams algorithm. The corresponding eigenvectors are obtained with an inverse iteration method using eigenfrequency-dependent trial eigenvectors. Residues of system harmonic transfer functions are computed using the eigenvalues and eigenvectors together with eigenvector normalization constants (modal Foss dampings). These residues are needed in a transient analysis. The method is applied to the generalized Beck column stability problem. Critical loads are calculated. Harmonic flexibilities of subcritically loaded columns in transverse vibration are studied. Applications to control problems are foreseen.