We summarize here our work on Lagrangian-stationary (aka H-minimal tori in C^2, i.e. stationary under Hamiltonian variations. On the one hand all tori can be described as finite type solutions for an integrable system. On the other hand, an explicit (and simple) parametrization is obtained using spinor-type representation and a Dirac equation. We explore the interplay between the two notions, and their relationship with a conjecture of Oh.
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