The problem of variables separation in the linear stability equations, which govern the disturbance behavior in viscous incompressible fluid flows, is considered. The so-called direct approach, in which a form of the 'Ansatz' for a solution with separated variables as well as a form of reduced ODEs are postulated from the beginning, is applied. The results of application of the method are the new coordinate systems and the most general forms of basic flows, which permit the postulated form of separation of variables. Thus, the stability analysis of nonparallel unsteady flows is reduced to the eigenvalue problems of ordinary differential equations. This method involves very complicated analytical calculations which can be implemented only using symbolic manipulating programs. The resulting eigenvalue problems are solved numerically with the help of the spectral collocation method based on Chebyshev polynomials. For some classes of perturbations, the eigenvalue problems can be solved analytically. Those unique examples of exact (explicit) solution of the nonparallel unsteady flow stability problems provide a very useful test for numerical methods of solution of eigenvalue problems, and for methods used in the hydrodynamic stability theory, in general.