  Catenoids in de Sitter $3$-space $S^3_1$ belong to a certain class of
space-like constant mean curvature one surfaces. In a previous work, the
authors classified such catenoids, and found that two different classes of
countably many exceptional elliptic catenoids are not realized as closed
subsets in $S^3_1$. Here we show that such exceptional catenoids have closed
analytic extensions in $S^3_1$ with interesting properties.
