Geometric aspects of Sturm-Liouville problems III. Level surfaces of the nth eigenvalue

For a given regular Sturm-Liouville equation and any natural number n, we explicitly describe the level surfaces of the nth eigenvalue of the equation, regarded as a function on the space of self-adjoint boundary conditions. The description of each level surface uses only a particular value of the transfer matrix of the equation. The shapes of these surfaces are also determined.