  We consider three Sturm--Liouville boundary value problems (the coercive ones
and the non-coercive one) in a bounded Lipschitz domain for the perturbed
Lam\'e operator with the boundary conditions of Robin type. We prove that the
problems are Fredholm ones in proper weighted Sobolev type spaces. The
conditions, providing the completeness of the root functions related to the
boundary value problem, are described.
