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We give a corrected statement of the theorem of Gurjar and Miyanishi, whichclassifies smooth affine surfaces of Kodaira dimension zero, whose coordinatering is factorial and has trivial units. Denote the class of such surfaces by$\mathcal{S}_{0}$. An infinite series of surfaces in $\mathcal{S}_{0}$, notlisted by Gurjar and Miyanishi, was recently obtained by Freudenburg, Kojimaand Nagamine as affine modifications of the plane. We complete their list to aseries containing arbitrarily high-dimensional families of pairwisenon-isomorphic surfaces in $\mathcal{S}_{0}$. Moreover, we classify them up toa diffeomorphism, showing that each occurs as an interior of a 4-manifold whoseboundary is an exceptional surgery on a 2-bridge knot. In particular, we showthat $\mathcal{S}_{0}$ contains countably many pairwise non-homeomorphicsurfaces.

Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units