We are concerned here with the transmission eigenvalues connected with a degerenate boundary problem involving a 2x2 system of differential operators. By a transmission eigenvalue of the boundary problem we mean a non-zero eigenvalue with the property that each component of a corresponding eigenvector is non-trivial. To derive information concerning the transmission eigenvalues, we proceed in the following way. Firstly we show that the Hilbert space operator induced by the boundary problem is just the linearization of a quadratic operator pencil which is parameter-elliptic. Then by utilizing some known facts concerning this pencil as well as some results pertaining to the uniqueness of the Cauchy problem, we are able to characterize the transmission eoigenvalues.