  The Neumann problem of linear elasticity is singular with a kernel formed by
the rigid motions of the body. There are several tricks that are commonly used
to obtain a non-singular linear system. However, they often cause reduced
accuracy or lead to poor convergence of the iterative solvers. In this paper,
different well-posed formulations of the problem are studied through
discretization by the finite element method, and preconditioning strategies
based on operator preconditioning are discussed. For each formulation we derive
preconditioners that are independent of the discretization parameter.
Preconditioners that are robust with respect to the first Lam\'e constant are
constructed for the pure displacement formulations, while a preconditioner that
is robust in both Lam\'e constants is constructed for the mixed formulation. It
is shown that, for convergence in the first Sobolev norm, it is crucial to
respect the orthogonality constraint derived from the continuous problem. Based
on this observation a modification to the conjugate gradient method is proposed
that achieves optimal error convergence of the computed solution.
