  For an arbitrary Hermitian period-$T$ Jacobi operator, we assume a
perturbation by a Wigner-von Neumann type potential to devise subordinate
solutions to the formal spectral equation for a (possibly infinite) real set,
$S$, of the spectral parameter. We employ discrete Levinson type techniques to
achieve this, which allow the analysis of the asymptotic behaviour of the
solution. This enables us to construct infinitely many spectral singularities
on the absolutely continuous spectrum of the periodic Jacobi operator, which
are stable with respect to an $l^1$-perturbation. An analogue of the
quantisation conditions from the continuous case appears, relating the
frequency of the oscillation of the potential to the quasi-momentum associated
with the purely periodic operator.
