  This paper studies a 2-class, 2-server parallel server system under the
recently introduced extended heavy traffic condition, which states that the
underlying 'static allocation' linear program (LP) is critical, but does not
require that it has a unique solution. The main result is the construction of
policies that asymptotically achieve a lower bound, proved in [1], on an
expected discounted linear combination of diffusion-scaled queue lengths, and
are therefore asymptotically optimal (AO).
  Each extreme point solution to the LP determines a control mode, i.e., a set
of activities (class--server pairs) that are operational.
  When there are multiple solutions, these modes can be selected dynamically.
It is shown that the number of modes required for AO is either one or two. In
the latter case there is a switching point in the (normalized) workload domain,
characterized in terms of a free boundary problem.
  Our policies are defined by identifying pairs of elementary policies and
switching between them at this switching point.
  They provide the first example in the heavy traffic literature where weak
limits under an AO policy are given by a diffusion process where both the drift
and diffusion coefficients are discontinuous.
