  We study the (weak) equilibrium problem arising from the problem of optimally
stopping a one-dimensional diffusion subject to an expectation constraint on
the time until stopping. The weak equilibrium problem is realized with a set of
randomized but purely state dependent stopping times as admissible strategies.
We derive a verification theorem and necessary conditions for equilibria, which
together basically characterize all equilibria. Furthermore, additional
structural properties of equilibria are obtained to feed a possible
guess-and-verify approach, which is then illustrated by an example.
