  In this paper we study a linearized eigenvalue problem derived from a a free
boundary problem modeling the growth of a tumor containing two species of
cells: proliferating cells and quiescent cells. The reduced form of this
eigenvalue problem is a $2$-system of a first-order nonlocal singular
differential-integral equation in a ball coupled by a third-order elliptic
pseudo-differential equation in the unit sphere. The singularity joined with
non-localness of the first-order equation causes the main difficulty of this
problem. By using Fourier expansion via a basis of spherical harmonic functions
and some techniques for solving singular differential integral equations
developed in some previous literature, we prove that there exists a null
sequence $\{\gamma_k\}_{k=2}^{\infty}$ for the surface tension coefficient
$\gamma$, with each of them being an eigenvalue of the linearized problem,
i.e., if $\gamma=\gamma_k$ for some $k\geq 2$ then the linearized problem has
extra nontrivial solutions besides the standard nontrivial solutions, and if
$\gamma\neq\gamma_k$ for all $k\geq 2$ then the linearized problem does not
have other nontrivial solutions than the standard nontrivial solutions.
Invertibility of some linear operators related to the linearized problem in
suitable function spaces is also studied.
