Considering the birefringence of the gain medium, we propose a selection rule for designing cavity configurations that enable the generation of radially and azimuthally polarized laser beams with a low beam quality factor, i.e., high beam quality. Using this rule, all stable regions can support cylindrical vector (CV) beams by using the position of the end mirror as the tuning parameter to vary the cavity configuration. Such cavity configurations tend to sustain the fundamental or lowest-order CV beam, and radially or azimuthally polarized beams can be obtained simply by varying the tuning parameter. Based on experimental measurements of the beam quality factor and polarization characteristics, we verified our analyses and simulations for a four-element laser system.