Smooth factorial affine surfaces of logarithmic Kodaira dimension zero with trivial units

This paper considers the family $\mathscr{S}_0$ of smooth affine factorial surfaces of logarithmic Kodaira dimension 0 with trivial units over an algebraically closed field $k$. Our main result (Theorem 4.1) is that the number of isomorphism classes represented in $\mathscr{S}_0$ is at least countably infinite. This contradicts the earlier classification of Gurjar and Miyanishi [5] which asserted that $\mathscr{S}_0$ has at most two elements up to isomorphism when $k=\mathbb{C}$. Thus, the classification of surfaces in $\mathscr{S}_0$ for the field $\mathbb{C}$, long thought to have been settled, is an open problem.