  The transmission eigenvalue problem is a system of two second-order elliptic
equations of two unknowns equipped with the Cauchy data on the boundary. In
this work, we establish the Weyl law for the eigenvalues and the completeness
of the generalized eigenfunctions for a system without complementing
conditions, i.e., the two equations of the system have the same coefficients
for the second order terms, and thus being degenerate. These coefficients are
allowed to be anisotropic and are assumed to be of class $C^2$. One of the keys
of the analysis is to establish the well-posedness and the regularity in
$L^p$-scale for such a system. As a result, we largely extend and rediscover
known results for which the coefficients for the second order terms are
required to be isotropic and of class $C^\infty$ using a new approach.
