Surfaces with pg = q = 1, K2 = 8 and nonbirational bicanonical map

We prove that if the bicanonical map of a minimal surface of general type S with pg = q = 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K_{S}^2=8}$$\end{document} is nonbirational, then it is a double cover onto a rational surface. An application of this theorem is the complete classification of minimal surfaces of general type with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p_{g}=q=1, K_{S}^2=8}$$\end{document} and nonbirational bicanonical map.