At turbulence saturation the decay of the medium and large scale motion (eddies, zonal flow, streamers in Tokamak plasmas) generates vortical structures. Their time of life and robustness against random perturbations determine the energy content at their specific scale. Examining analytical models of slowly evolving structures we find that both quasi-integrable and exactly integrable vortices are possible. At space scales larger than the ion Larmor radius and in quasi-ideal plasma the model equation for stationary strongly nonlinear ion modes may take the form of Jacobs-Rebbi and Flierl-Petviashvili equations. The Jacobs-Rebbi equation (also known in superfluid physics) has vortex solutions that have been obtained only numerically. We prove that this equation is exactly integrable, finding the Lax pair of operators and constructing the spectral Riemann surface that provides systematic solutions on periodic domains. The analytical expression for exact solutions of this equation is provided, in therms of the Riemann Theta functions whose arguments are determined by the boundary conditions. We also provide explicit examples of derivation of the solutions for some generic potential distributions relevant for the Tokamak plasma.

By contrast, the monopolar two-dimensional vortex of the Flierl-Petviashvili equation is not an integrable structure which explains the faster decay by drift wave radiation. The quasi-stability of this solution is due to the proximity in function spaces from exact structures like those generated by Liouville equation and sinhyperbolic equation.

We present a practical procedure which allows to check if a potential structure (obtained from experimental measurements or from numerical simulations) corresponds to a solution of a particular nonlinear equation. The method permits to characterize the state to which evolves the plasma at saturated turbulence, associating the evolution in the stationary state with a nonlinear model. This method is more detailed than the description in terms of statistical quantities.