  We consider a quadratic operator pencil with a small periodic perturbation
multiplied by the spectral parameter. It is motivated, in particular, by a
one-dimensional Klein-Gordon equation with a time-parity-symmetric
perturbation. We study in details the structure of the considered operator
pencil. We show that its essential spectrum has a band structure and at certain
thresholds, the bands bifurcate into small parabolas. We then study how the
isolated limiting eigenvalues behave under the perturbation. We show that if
zero is a limiting isolated eigenvalue, under the perturbation it remains an
eigenvalue but an additional isolated eigenvalue can emerge from zero. The most
part of the paper is devoted to studying the isolated eigenvalues converging to
the essential spectrum. We establish sufficient conditions for the existence
and absence of such eigenvalues and in the case of the existence, we calculate
the leading terms of their asymptotic expansions.
