We investigate finite elation Laguerre planes admitting a group of automorphisms
that is two-transitive on the set of generators. We exclude the sporadic cases of
socles in two-transitive groups, as well as the alternating and Suzuki groups
and the cases with abelian socle (except for the smallest ones, where the
Laguerre planes are Miquelian of order at most four). The remaining cases
are dealt with in a separate paper. We prove that a finite elation Laguerre
plane is Miquelian if its automorphism group is two-transitive on the set
of generators. Equivalently, each translation generalized quadrangle of
order
with a group of automorphisms acting two-transitively on the set of lines through the
base point is classical.
