  This article determines the spectral data, in the integrable systems sense,
for all weakly conformally immersed Hamiltonian stationary Lagrangian in
$\R^4$. This enables us to describe their moduli space and the locus of branch
points of such an immersion. This is also an informative example in integrable
systems geometry, since the group of ambient isometries acts non-trivially on
the spectral data and the relevant energy functional (the area) need not be
constant under deformations by higher flows.
