  It is a common assumption that quantum systems with time reversal invariance
and classically chaotic dynamics have energy spectra distributed according to
GOE-type of statistics. Here we present a class of systems which fail to follow
this rule. We show that for convex billiards of constant width with time
reversal symmetry and "almost" chaotic dynamics the energy level distribution
is of GUE-type. The effect is due to the lack of ergodicity in the "momentum"
part of the phase space and, as we argue, is generic in two dimensions.
Besides, we show that certain billiards of constant width in multiply connected
domains are of interest in relation to the quantum ergodicity problem. These
billiards are quantum ergodic, but not classically ergodic.
