At space scales of the order of the ion Larmor radius the strongly nonlinear structures are described by the Hasegawa-Mima equation. The stationary form of this equation is very general and only provides a weak constraint on the space of solutions. This suggests that the stationary solutions must be identified by a different approach. We first note that the localized structures are in general associated in a two-dimensional geometry with point-like vortices interacting via a short-range potential (in contrast to the ideal Euler fluid, where the interaction is logarithmic). We develop a detailed model of dynamical system of discrete vortices with interaction potential derived from both Maxwell and Chern-Simons Lagrangeans. The continuous limit of this discrete model is a field theory leading, at the saturation of an inequality obeyed by the action functional, to an equation describing the stationary states of the Hasegawa-Mima equation. This equation is similar with the sin-hyperbolic Poisson equation for Euler fluids, but is not conformally invariant.
We prove that this equation is exactly integrable and provide the explicit form of the Lax operators. We then develop in detail the method of finding the exact analytical solutions. For a concrete example we assume a particular choice of boundary conditions and calculate explicitely the parameters of the solution and the solution itself, in terms of Riemann theta functions. The problem of stability of these solution is discussed, on the basis of the change, due to some perturbation, of the topology of the spectral hyperelliptic surface. Connection with the measurements of the electrostatic potential and exploitation of the analytical integrability in experiments are presented.