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Non-algebraic curves play an increasing role in string theory, sometimes they are known to be related to the integrable systems of the KP/Toda type. Are there any investigated examples of the application of the Krichever-Novikov-Dubrovin description of integrable hierarchies to non-algebraic spectral curves?

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There definitely have been attempts to generalize Krichever formula for algebro-geometric solutions of KP equation to the case of infinite genus spectral curves. For example, see Feldman--Knoerrer-Trubowitz's Infinite Genus Riemann Surfaces or McKean--Trubowitz's HILL'S SURFACES AND THEIR THETA FUNCTIONS.

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