  We consider the integrable one-dimensional spin-$1$ chain defined by the
Zamolodchikov-Fateev (ZF) Hamiltonian. The latter is parametrized, analogously
to the XXZ spin-$1/2$ model, by a continuous anisotropy parameter and at the
isotropic point coincides with the well-known spin-$1$ Babujian-Takhtajan
Hamiltonian. Following a procedure recently developed for the XXZ model, we
explicitly construct a continuous family of quasi-local conserved operators for
the periodic spin-$1$ ZF chain. Our construction is valid for a dense set of
commensurate values of the anisotropy parameter in the gapless regime where the
isotropic point is excluded. Using the Mazur inequality, we show that, as for
the XXZ model, these quasi-local charges are enough to prove that the
high-temperature spin Drude weight is non-vanishing in the thermodynamic limit,
thus establishing ballistic spin transport at high temperature.
