  Partially rectangular domains are compact two-dimensional Riemannian
manifolds $X$, either closed or with boundary, that contain a flat rectangle or
cylinder. In this paper we are interested in partially rectangular domains with
ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard
or Donnelly surfaces.
  We consider a one-parameter family $X_t$ of such domains parametrized by the
aspect ratio $t$ of their rectangular part. There is convincing theoretical and
numerical evidence that the Laplacian on $X_t$ with Dirichlet or Neumann
boundary conditions is not quantum unique ergodic (QUE). We prove that this is
true for all $t \in [1,2]$ excluding, possibly, a set of Lebesgue measure zero.
This yields the first examples of ergodic billiard systems proven to be
non-QUE.
