The classification of Hamiltonian stationary Lagrangian tori in {{mathbb {CP}}^2} by their spectral data

It is known that all weakly conformal Hamiltonian stationary Lagrangian immersions of tori in ${{mathbb {CP}}^2}$ may be constructed by methods from integrable systems theory. This article describes the precise details of a construction which leads to a form of classification. The immersion is encoded as spectral data in a similar manner to the case of minimal Lagrangian tori in ${{mathbb {CP}}^2}$ , but the details require a careful treatment of both the ??dressing construction?? and the spectral data to deal with a loop of flat connexions which is quadratic in the loop parameter.