Degenerate Boundary Conditions for a Third-Order Differential Equation

We consider the spectral problem y'''(x) = λy(x) with general two-point boundary conditions that do not contain the spectral parameter λ. We prove that the boundary conditions in this problem are degenerate if and only if their 3 × 6 coefficient matrix can be reduced by a linear row transformation to a matrix consisting of two diagonal 3 × 3 matrices one of which is the identity matrix and the diagonal entries of the other are all cubic roots of some number. Further, the characteristic determinant of the problem is identically zero if and only if that number is −1. We also show that the problem in question cannot have finite spectrum.