Explore quantum integrable systems dynamics through coupled Heisenberg equations in this 30-minute conference talk by Oleg Lychkovskiy. Delve into the analytical approach for solving Heisenberg equations in integrable systems, focusing on models where the Hamiltonian is an element of the Onsager algebra and the Kitaev model on the honeycomb-like Bethe lattice. Discover a new algebra containing the Hamiltonian and learn about analytical results for quench dynamics unavailable through other methods. Gain insights into the prospects of this approach, related open problems, and how advancements in understanding underlying algebras can lead to new discoveries in quench dynamics. The talk covers topics such as quantum dynamics of observables, Heisenberg representation, transverse-field Ising model, Onsager algebra in nearest-neighbor spin models, and the Kitaev model on a honeycomb lattice.