The model two-level problem of non-LTE line formation in homogeneous plane atmospheres is reconsidered with the complete account of polarization arising in resonance scattering. We use the approximation of complete frequency redistribution (CFR) and restrict our discussion to the most important case of axially symmetric radiation fields in semi-infinite atmospheres. The primary sources are assumed to be partially polarized. The problem is reduced to the 2 × 2 matrix Wiener-Hopf integral equation for the matrix source function S(τ). The matrix kernel K₁(τ) of the Λ-operator appearing in this equation is represented as a continuous superposition of exponentials. As we show in Paper II of the series, this enables one to develop a matrix version of the analytical theory which, on the one hand, is a generalization of the scalar CFR theory and, on the other, is the CFR version of the theory of multiple monochromatic Rayleigh scattering. As a preparatory step for this, we discuss in detail the properties of the kernel matrix K₁(τ) and the dispersion matrix T(z). The latter is essentially the two-sided Laplace transform of K₁(τ). We consider the asymptotic behavior of K₁(τ) and T(z) for large τ and z, respectively. For the particular case of the Doppler profile the complete asymptotic expansions of these matrices are presented. These results are at the base of the theory presented in Paper II of the series.