General solutions of the maser polarization problem are presented for arbitrary absorption coefficients. The results are used to calculate polarization for masers permeated by magnetic fields with arbitrary values of x(B), the ratio of Zeeman splitting to Doppler linewidth, and for anisotropic (m-dependent) pumping. In the case of magnetic fields, one solution describes the polarization for overlapping Zeeman components, x(B) < 1. The x(B) --> 0 limit of this solution reproduces the linear polarization derived in previous studies, which were always conducted at this unphysical limit. Terms of higher order in x(B) have a negligible effect on the magnitude of q. However, these terms produce some major new results. (1) The solution is realized only when the Zeeman splitting is sufficiently large that x(B) > (S-0/J(s))(1/2), where S-0 is the source function and J(s) is the saturation intensity (pumping schemes typically have S-0/J(s) similar to 10(-5) to 10(-8)). When this condition is met, the linear polarization requires J/J(s) greater than or similar to x(B), where J is the angle-averaged intensity. This condition generally requires considerable amplification, but is met long before saturation (J/J(s) greater than or equal to 1). (2) The linear polarization is accompanied by circular polarization, proportional to x(B). Because x(B) is proportional to the transition wavelength, the circular polarization of SiO masers should decrease with rotation quantum number, as observed. In the absence of theory for x(B) < 1, previous estimates of magnetic fields from detected maser circular polarization had to rely on conjectures in this case and generally need to be revised downward. The fields in SiO masers are similar to 2-10 G and were overestimated by a factor of 8. The OH maser regions around supergiants have fields of similar to 0.1-0.5 mG, which were overestimated by factors of 10-100. The fields were properly estimated for OH/IR masers (less than or similar to 0.1 mG) and H2O masers in star-forming regions (similar to 15-50 mG). (3) Spurious solutions that required stability analysis for their removal in all previous studies are never reproduced here; in particular, there are no stationary physical solutions for propagation at sin(2) theta < 1/3, where theta is the angle from the direction of the magnetic field, so such radiation is unpolarized. These spurious solutions can be identified as the x(B) = 0 limits of nonphysical solutions and they never arise a finite values of x(B), however small. (4) Allowed values of theta are limited by bounds that depend both on Zeeman splitting and frequency shift from line center. At x(B) less than or similar to 10(-3), the allowed phase space region encompasses essentially all frequenceis and sin(2) theta > 1/3. As the field strength increases, the allowed angular region shrinks at a frequency-dependent rate, leading to contraction of the allowed spectral region. This can result in narrow maser features with linewidths smaller than the Doppler width and substantial circular polarization in sources with x(B) greater than or similar to 0.1. When x(B) greater than or equal to 0.7, all frequencies and directions are prohibited for the stationary solution and the radiation is unpolarized. Another solution describes the polarization when the Zeeman components separate. This occurs at line center when x(B) > 1 and at one Doppler width when x(B) > 2. The solution is identical to that previously identified in the x(B) --> infinity limit, and applies to OH masers around H II regions. A significant new result involves the substantial differences between the pi- and sigma-components for most propagation directions, differences that persist into the saturated domain. Overall, H II/OH regions should display a preponderance of sigma-components. The absence of any pi-components in W3(OH) finds a simple explanation as maser action in a magnetic field aligned within sin(2) theta < 2/3 to the line of sight.
