  We consider the first Weyl algebra, A, in the Euler gradation, and completely
classify graded rings B that are graded equivalent to A: that is, the
categories gr-A and gr-B are equivalent. This includes some surprising
examples: in particular, we show that A is graded equivalent to an idealizer in
a localization of A. We obtain this classification as an application of a
general Morita-type characterization of equivalences of graded module
categories.
