Suppose we have a quadratic eigenvalue problem $(A_{0}+\lambda A_{1}+ \lambda^{2} A_{2})x=0$. I'd to know if there are conditions under which the problem is known to have a small number of *distinct* eigenvalues (say, 3 or 4). If it helps, I can assume that $A_{0}=I,A_{1}=I-J$ and that $A_{2}$ is negative semidefinite.

P.S. $J$ is the all-ones matrix.