  In this paper we study one-dimensional quantum Ising spin chains in external
magnetic field close to an integrable point. We concentrate on the dynamics of
the slowest operator, that plays a key role at the final period of
thermalization. We introduce two independent definitions of the slowest
operator: local and translationally-invariant ones. We construct both operators
numerically using tensor networks and extensively compare their physical
properties. We find that the local operator has a significant overlap with
energy flux, it does not correspond to an integral of motion, and, as one goes
away from the integrable point, its revivals get suppressed and the rate of
delocalization changes from extremely slow to slower than diffusion. The
translationally-invariant operator corresponds to an integral of motion; as the
system becomes less integrable, at some point this operator changes its nature:
from no overlap with any magnetization and fast rate of delocalization, to
non-zero overlap with magnetizations $\sigma_{x}$ and $\sigma_{z}$ and slow
rate of delocalization.
