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Comic historian Steve Holland wrote a short but interesting article about this for a German Don Lawrence fanzine (for those interested, see: "Pandarve - Don Lawrence Magazin", 1991, issue 3, page 21), in which he explained Don's contribution to Billy The Kid.
According to Steve Holland, the first Billy The Kid episode drawn by Don Lawrence was published in Sun #491 (July 5, 1958), and Don would continue to draw 16 more stories.
But perhaps he even made more, it is not known. Some of these stories (if not all) were written by Michael Butterworth, long before he worked together with Don Lawrence on "The Rise and Fall of the Trigan Empire" (1965). So far, none of these stories have been reprinted in modern times, which is quite surprising.
Unfortunately, the Billy The Kid SUN Collections on this website only run until December 14, 1957, long before Don Lawrence started working on this series.