  We consider the boundary value problem for the deflection of a finite beam on
an elastic foundation subject to vertical loading. We construct a one-to-one
correspondence $\Gamma$ from the set of equivalent well-posed two-point
boundary conditions to $\mathrm{gl}(4,\mathbb{C})$. Using $\Gamma$, we derive
eigenconditions for the integral operator $\mathcal{K}_\mathbf{M}$ for each
well-posed two-point boundary condition represented by $\mathbf{M} \in
\mathrm{gl}(4,8,\mathbb{C})$. Special features of our eigenconditions include;
(1) they isolate the effect of the boundary condition $\mathbf{M}$ on
$\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$, (2) they connect
$\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$ to
$\mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$ whose structure has been well
understood. Using our eigenconditions, we show that, for each nonzero real
$\lambda \not \in \mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$, there exists a real
well-posed boundary condition $\mathbf{M}$ such that $\lambda \in
\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$. This in particular shows that the
integral operators $\mathcal{K}_\mathbf{M}$ arising from well-posed boundary
conditions, may not be positive nor contractive in general, as opposed to
$\mathcal{K}_{l,\alpha,k}$.
