We present a new approach, based on Richards-Wolf formalism, to rigorously model nonparaxial focusing of radially and azimuthally polarized electromagnetic beams by axisymmetric systems without a single-point focus. Our approach is based on a combined method that uses ray tracing and diffraction integrals. Our method is validated by comparing known results obtained with a parabolic mirror. Our integral representation of the focused beams, compliant with diffraction theory, is thoroughly discussed and solved for various conics that, so far, have not been treated analytically. The extension of the method to other polarization states is straightforward.