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