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We study the space of periodic solutions of the elliptic $\sinh$-Gordonequation by means of spectral data consisting of a Riemann surface $Y$ and adivisor $D$. We show that the space $M_g^{\mathbf{p}}$ of real periodic finitetype solutions with fixed period $\mathbf{p}$ can be considered as a completelyintegrable system $(M_g^{\mathbf{p}},\Omega,H_2)$ with a symplectic form$\Omega$ and a series of commuting Hamiltonians $(H_n)_{n \in \mathbb{N}}$. Inparticular we relate the gradients of these Hamiltonians to the Jacobi fields$(\omega_n)_{n\in \mathbb{N}_0}$ from the Pinkall-Sterling iteration. Moreover,a connection between the symplectic form $\Omega$ and Serre duality isestablished.

Periodic solutions of the sinh-Gordon equation and integrable systems