Action–angle variables and conservation laws expressed in terms of scattering data for an integrable hierarchy associated with the Zakharov–Ito system

In this work, the Poisson brackets for the scattering data of an integrable hierarchy for the Zakharov–Ito system are computed by employ the inverse scattering approach. Then, the action–angle variables and infinite conservation laws are expressed in terms of the scattering data. It is worth noting that, compared to previous works on Camassa–Holm equation and KdV-type equations, we have made some further progress. We have, for the first time, obtained action–angle variables that holds for the entire integrable hierarchy associated with the Zakharov–Ito system. Furthermore, we prove that the variational derivative of action–angle variables are eigenfunctions of the recursive operator, thus obtaining an infinite set of conservation laws in terms of scattering data.