We study the asymptotic behavior of solutions to the linear problem of elasticity in a domain Ω with paraboloidal exit at infinity. Properties of solutions and the condition of the existence of solutions depend on a parameter γ ∈ [0,1] characterizing the velocity of extending the paraboloid (a cylinder and a cone correspond to the cases γ = 0 γ = 1 respectively). Asymptotic formulas are deduced for displacement fields generating forces and moments "applied at infinity." The Saint-Venant principle is verified for "oblong" bodies such as paraboloids, cylinders, and narrow cones. The following question turns out to be a key one: What rigid displacements belong to the energy space obtained by completion of C ₀ ^∞ (Ω̄) ³ by the energy norm? The dimension d _γ of the lineal R _γ of rigid energy displacements is computed (in this case, d ₀ = 6, d ₁ = 0, and the function γ → d _γ has jumps at the points γ = 1/4, 1/2, 3/4). We also clarify the reasons why it is necessary to distinguish the notions "energy solution" and "solution with finite energy." We also discuss the phenomenon of a boundary layer that appears near the endpoints of spindle-like rods and is described by energy solutions in paraboloids. As is shown, in order to have the well-posed formulation of the boundary conditions in one-dimensional models of such rods, it is necessary to use the weakened Saint-Venant principle, i.e., replace R ₀ with R _γ : for γ > 1/4. If we apply the strong principle, we arrive at an overdetermined limit one-dimensional problem.