A method for solving convolution type equations (in particular, Wiener-Hopf equations) on the basis of the Volterra factorization of the integral operators and the analysis of the nonlinear functional factorization equations was suggested in \textit{L. G. Arabadzhyan} and \textit{N. B. Engibaryan}'s [Itogi Nauki Tekh. Mat. Anal. 22, 175--244 (1984; Zbl 0568.45004)]. In this paper, the author investigates the factorization of conservative integral convolution type operators with slowly decaying kernels and proves a theorem that contains Theorems 1 and 2 in his paper [Mat. Zametki 46, No.~1, 3--10 (1989; Zbl 0724.45002)] as special cases. In particular, he shows that if the kernel \(K\) of the homogeneous Wiener-Hopf integral equation \[ S(x) =\int_0^\infty K(x-t) S(t) dt , \quad x\in [0, +\infty), \] be conservative, then this equation has a positive solution \(S\) monotone increasing on \([0, +\infty )\).