The author determines the structure of all Lie groups acting transitively on the direct product of the circle and an even-dimensional sphere. For products of two spheres of dimension \(>1\) similar problems were solved by other authors. The author also determines the minimal transitive Lie groups on \(S^1\times S^{2m}\). The case of the direct product of \(S^1\) with an odd-dimensional sphere is not considered.
As an application, and a motivation for the paper, the structure of the automorphism group of a class of generalized quadrangles, special Tits buildings, is considered. A conjecture of L. Kramer [Mem. Am. Math. Soc. 752, 113 p. (2002; Zbl 1027.57002)] and O. Bletz-Siebert [J. Lie Theory 15, No. 1, 1–11 (2005; Zbl 1067.57033)] is proved saying that the automorphism group of a connected generalized quadrangle of type \((1,2m)\) always contains a transitive subgroup that is the direct product of a compact simple Lie group and a one-dimensional Lie group. It is conjectured that the case of type \((1,2 m+1)\) is similar.