  Exceptional domains are domains on which there exists a positive harmonic
function, zero on the boundary and such that the normal derivative on the
boundary is constant. Recent results classify exceptional domains as belonging
to either a certain one-parameter family of simply periodic domains or one of
its scaling limits.
  We introduce quasi-exceptional domains by allowing the boundary values to be
different constants on each boundary component. This relaxed definition retains
the interesting property of being an \emph{arclength quadrature domain}, and
also preserves the connection to the hollow vortex problem in fluid dynamics.
We give a partial classification of such domains in terms of certain Abelian
differentials. We also provide a new two-parameter family of periodic
quasi-exceptional domains. These examples generalize the hollow vortex array
found by Baker, Saffman, and Sheffield (1976). A degeneration of regions of
this family provide doubly-connected examples.
