We give a corrected statement of (Gurjar and Miyanishi 1988, Theorem 2), which
classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate
ring is factorial and has trivial units. Denote the class of such surfaces by
. An infinite series
of surfaces in
,
not listed in loc. cit., was recently obtained by Freudenburg, Kojima and Nagamine
(2019) as affine modifications of the plane. We complete their list to a series
containing arbitrarily high-dimensional families of pairwise nonisomorphic surfaces in
. Moreover,
we classify them up to a diffeomorphism, showing that each occurs as an interior of a
-manifold
whose boundary is an exceptional surgery on a
-bridge knot. In
particular, we show that
contains countably many pairwise nonhomeomorphic surfaces.
