  We conjecture the existence of hidden Onsager algebra symmetries in two
interacting quantum integrable lattice models, i.e. spin-1/2 XXZ model and
spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the
anisotropy. The conjectures relate the Onsager generators to the conserved
charges obtained from semi-cyclic transfer matrices. The conjectures are
motivated by two examples which are spin-1/2 XX model and spin-1 U(1)-invariant
clock model. A novel construction of the semi-cyclic transfer matrices of
spin-1 Zamolodchikov-Fateev model at arbitrary root of unity value of the
anisotropy is carried out via transfer matrix fusion procedure.
