  We investigate the possibility to determine the divergence-free displacement
$\mathbf{u}$ \emph{independently} from the pressure reaction $p$ for a class of
boundary value problems in incompressible linear elasticity. If not possible,
we investigate if it is possible to determine it \emph{pressure robustly}, i.e.
pollution free from the pressure reaction.
  For convex domains there is but one variational boundary value problem among
the investigated that allows the independent determination. It is the one with
essential no-penetration conditions combined with homogeneous tangential
traction conditions.
  Further, in most but not all investigated cases, the weakly divergence-free
displacement can be computed pressure robustly provided the total body force is
decomposed into its direct sum of divergence- and rotation-free components
using a Helmholtz decomposition. The elasticity problem is solved using these
components as separate right-hand sides. The total solution is obtained using
the superposition principle.
  We employ a $(\mathbf{u},p)$ higher-order finite element formulation with
discontinuous pressure elements. It is \emph{inf-sup} stable for polynomial
degree $p\ge 2$ but not pressure robust by itself.
  We propose a three step procedure to solve the elasticity problem preceded by
the Helmholtz decomposition of the total body force. The extra cost for the
three-step procedure is essentially the cost for the Helmholtz decomposition of
the assembled total body force, and the small cost of solving the elasticity
problem with one extra right-hand side. The results are corroborated by
theoretical derivations as well as numerical results.
