  Recently found spin-flip non-invariant (SFNI) conserved quantities play
important roles in discussing nonequilibrium physics of the XXZ model. The
representative examples are the generalized Gibbs ensemble (GGE) and the
ballistic transport of the spin current. In spite of big progress in
understanding nonequilibrium physics of integrable systems, the general
framework to determine a minimal complete set of conserved quantities which
describes the long-time steady state has not yet been found. This paper shows
that the GGE of the gapless XXZ model consists of functionally independent
conserved quantities rather than linearly independent. At the same time, the
physical meaning of SFNI conserved quantities is provided. We also discuss that
there exist ballistic channels of the spin current supported by non-quasilocal
conserved quantities. The saturation of the lower bound for the Drude weight by
quasilocal conserved quantities reads the linear dependence of non-quasilocal
conserved quantities on quasilocal ones. We show that their (generalized)
linearly dependence relation is consistent with the statement that the GGE
consists of functionally independent conserved quantities without containing
all linearly independent conserved quantities.
