  One of the first and therefore most important theorems in perturbation theory
claims that for an arbitrary self-adjoint operator A there exists a
perturbation B of Hilbert-Schmidt class with arbitrary small operator norm,
which destroys completely the absolutely continuos (a.c.) spectrum of the
initial operator A (von Neumann). However, if A is the discrete free 1-D
Schr\"odinger operator and B is an arbitrary Jacobi matrix (of Hilbert-Schmidt
class) the a.c. spectrum remains perfectly the same, that is, the interval
[-2,2]. Moreover, Killip and Simon described explicitly the spectral properties
for such A+B. Jointly with Damanik they generalized this result to the case of
perturbations of periodic Jacobi matrices in the non-degenerated case. Recall
that the spectrum of a periodic Jacobi matrix is a system of intervals of a
very specific nature. Christiansen, Simon and Zinchenko posed in a review
dedicated to F. Gesztesy (2013) the following question: "is there an extension
of the Damanik-Killip-Simon theorem to the general finite system of intervals
case?" In this paper we solve this problem completely. Our method deals with
the Jacobi flow on GSMP matrices. GSMP matrices are probably a new object in
the spectral theory. They form a certain Generalization of matrices related to
the Strong Moment Problem, the latter ones are a very close relative of Jacobi
and CMV matrices. The Jacobi flow on them is also a probably new member of the
rich family of integrable systems. Finally, related to Jacobi matrices of
Killip-Simon class, analytic vector bundles and their curvature play a certain
role in our construction and, at least on the level of ideology, this role is
quite essential.
