  It is known that all weakly conformal Hamiltonian stationary Lagrangian
immersions of tori in the complex projective plane 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 the complex projective plane, 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.
