This article considers the following. Let \(\pi\) be a finite translation plane of order \(p^ r\) with an autotopism group G which has an orbit of length \(p^ r\)-p on \(\ell_{\infty}\), the line at infinity. The authors make the following additional assumptions: (a) p is an odd prime and \(r=2\); (b) G acts faithfully on \(\ell_{\infty}\). The aim is to give several examples. First, they show that the group G must leave invariant a subplane of order p or \(\pi\) is a Hall plane. They make the additional assumption: (c) G leaves invariant a subplane \(\pi_ 0\) of \(\pi\) which without loss of generality can be taken to be the kern subplane. They then prove that G has a unique Sylow p-subgroup. It follows that the plane \(\pi\) is a \(\Delta\)-transitive plane in the sense of \textit{V. Jha} [Arch. Math. 37, 377-384 (1981; Zbl 0455.51003)] who conjectured that the Hall planes of order \(p^ 2\) are the only \(\Delta\)-transitive planes. As already indicated most of the article deals with constructing examples of the planes \(\pi\). They give constructions for \(p\equiv -1\)(mod 6), for \(p\equiv 2\)(mod 5), and for \(p\equiv 3\)(mod 5). They then show how to extend these constructions to the case when p is an odd prime power. In particular they show that the translation planes derived from weak nucleus semifield planes, where the left nucleus is the weak nucleus, admit an autotopism group fixing the keen subplane \(\pi_ 0\) and acting transitively on \(\ell_{\infty}-(\ell_{\infty}\cap \pi_ 0).\)