Transitive Lie groups on $S^1\times S^{2m}$

The structure of Lie groups acting transitively on the direct product of a circle and an even-dimensional sphere is described. For products of two spheres of dimension $>1$ a similar problem has already been solved by other authors. The minimal transitive Lie groups on $S^1$ and $S^{2m}$ are also indicated.
As an application of these results, the structure of the automorphism group of one class of geometric structures, generalized quadrangles (a special case of Tits buildings) is considered. A conjecture put forward by Kramer is proved: 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.
Bibliography: 16 titles.