  We study the class of hyponormal 2-variable weighted shifts with two
consecutive equal weights in the weight sequence of one of the coordinate
operators. We show that under natural assumptions on the coordinate operators,
the presence of consecutive equal weights leads to horizontal or vertical
flatness, in a way that resembles the situation for 1-variable weighted shifts.
In 1-variable, it is well known that flat weighted shifts are necessarily
subnormal (with finitely atomic Berger measures). By contrast, we exhibit a
large collection of flat (i.e., horizontally and vertically flat) 2-variable
weighted shifts which are hyponormal but not subnormal. Moreover we completely
characterize the hyponormality and subnormality of symmetrically flat
contractive 2-variable weighted shifts.
